A KOSZUL DUALITY for PROPS Introduction the Koszul Duality Is a Theory Developed for the First Time in 1970 by S. Priddy For

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 10, October 2007, Pages 4865–4943 S 0002-9947(07)04182-7 Article electronically published on May 16, 2007

AKOSZULDUALITYFORPROPS

BRUNO VALLETTE

Abstract. The notion of prop models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props.

Introduction The Koszul duality is a theory developed for the ﬁrst time in 1970 by S. Priddy for associative algebras [Pr]. To every quadratic algebra A, it associates a dual coalgebra A¡ and a chain complex called a Koszul complex. When this complex is acyclic, we say that A is a Koszul algebra. In this case, the algebra A and its representations have many properties (cf. A. Beilinson, V. Ginzburg and W. Soergel [BGS]). In 1994, this theory was generalized to algebraic operads by V. Ginzburg and M.M. Kapranov (cf. [GK]). An operad is an algebraic object that models the operations with n inputs (and one output) A⊗n → A acting on a type of algebras. For instance, there exist operads As, Com and Lie coding associative, commutative and Lie algebras. The Koszul duality theory for operads has many applications: construction of a “small” chain complex to compute the homology groups of an algebra, minimal model of an operad, and notion of algebra up to homotopy. The discovery of quantum groups (cf. V. Drinfeld [Dr1, Dr2]) has popularized algebraic structures with products and coproducts, that’s to say, operations with multiple inputs and multiple outputs A⊗n → A⊗m . It is the case of bialgebras, Hopf algebras and Lie bialgebras, for instance. The framework of operads is too narrow to treat such structures. To model the operations with multiple inputs and outputs, one has to use a more general algebraic object: the props. Following J.-P. Serre in [S], we call an “algebra” over a prop P,aP-algebra. It is natural to try to generalize Koszul duality theory to props. A few works have been done in that direction by M. Markl and A.A. Voronov in [MV] and W.L. Gan in [G]. Actually, M. Markl and A.A. Voronov proved Koszul duality theory 1 for what M. Kontsevich calls 2 -PROPs, and W.L. Gan proved it for dioperads. One has to bear in mind that an associative algebra is an operad, an operad is a 1 1 2 -PROP, a 2 -PROP is a dioperad and a dioperad induces a prop: 1 Associative algebras → Operads → − PROPs → Dioperads ; Props. 2

Received by the editors June 27, 2005. 2000 Mathematics Subject Classiﬁcation. Primary 18D50; Secondary 16W30, 17B26, 55P48. Key words and phrases. Prop, Koszul duality, operad, Lie bialgebra, Frobenius algebra.

c 2007 American Mathematical Society Reverts to public domain 28 years from publication 4865

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In this article, we introduce the notion of properad, which is the connected part of a prop and contains all the information to code algebraic structure like bialgebras, Lie bialgebras, and infinitesimal Hopf algebras (cf. M. Aguiar [Ag1], [Ag2], [Ag3]). We prove the Koszul duality theory for properads which gives Koszul duality for quadratic props defined by connected relations (it is the case of the two last examples). The first difficulty is to generalize the bar and cobar constructions to properads and props. The differentials of the bar and cobar constructions for associative 1 algebras, operads, 2 -PROPs and dioperads are given by the notions of ‘edge contraction’ and ‘vertex expansion’ introduced by M. Kontsevich in the context of graph homology (cf. [Ko]). We define the differentials of the bar and cobar constructions of properads using conceptual properties. This generalizes the previous constructions. The preceding proofs of Koszul duality theories are based on combinatorial properties of trees or graphs of genus 0. Such proofs cannot be used in the context of props since we have to work with any kind of graph. To solve this problem, we introduce an extra grading induced by analytic functors and generalize the comparison lemmas of B. Fresse [Fr] to properads. To any quadratic properad P, we associated a Koszul dual coproperad P¡ and a Koszul complex. The main theorem of this paper is a criterion which claims that the Koszul complex is acyclic if and only if the cobar construction of the Koszul dual P¡ is a resolution of P. In this case, we say that the properad is Koszul and this resolution is the minimal model of P. As a corollary, we get the deformation theories for gebras over Koszul properads. For instance, we prove that the properad of Lie bialgebras and the properad of infinitesimal Hopf algebras are Koszul, which gives the notions of Lie bialgebra up to homotopy and infinitesimal Hopf algebra up to homotopy. Structures of gebras up to homotopy appear in the level of on some chain complexes. For instance, the Hochschild cohomology of an associative algebra with coefficients into itself has a structure of Gerstenhaber algebra up to homotopy (cf. J.E. McClure and J.H. Smith [McCS], see also C. Berger and B. Fresse [BF]). This conjecture is known as Deligne’s conjecture. The same kind of conjectures exist for gebras with coproducts up to homotopy. One of them was formulated by M. Chas and D. Sullivan in the context of String Topology (cf. [CS]). This conjecture says that the structure of involutive Lie bialgebra (a particular type of Lie bialgebras) on some homology groups can be lifted to a structure of involutive Lie bialgebras up to homotopy. Working with complexes of graphs of genus greater than 0 is not an easy task. This Koszul duality for props provides new methods for studying the homology of these chain complexes. One can interpret the bar and cobar constructions of Koszul properads in terms of graph complexes. Therefore, it is possible to compute their (co)homology in Kontsevich’s sense. For instance, the case of the properad of Lie bialgebras leads to the computation of the cohomology of classical graphs, and the case of the properad of infinitesimal Hopf algebras leads to the computation of the cohomology of ribbon graphs (cf. M. Markl and A.A. Voronov [MV]). In these cases, the homology groups are given by the associated Koszul dual coproperad. So we get a complete description of them as S-bimodules. In a different

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context, S. Merkulov recently gave another proof of the famous formality theorem of M. Kontsevich using the properties of chain complexes of Koszul props in [Me]. In the last section of this paper, we generalize the Poincaré series of associative algebras and operads to properads. When a properad P is Koszul, we prove a functional equation between the PoincaréseriesofP and the Poincaréseriesofits Koszul dual P¡.

Conventions Throughout the text, we will use the following conventions. We work over a ﬁeld k of characteristic 0.

0.1. n-tuples. To simplify the notations, we often write ¯ı an n-tuple (i1, ..., in). The n-tuples considered here are n-tuples of non-negative integers. We denote the sum i1 + ···+ in by |¯ı|. When there is no ambiguity about the number of terms, the n-tuple (1, ..., 1) is denoted by 1.¯ We use the notation ¯ı to represent the “products” of elements indexed by the n-tuple (i1, ..., in). For instance, in the case of k-modules, V¯ı corresponds to ⊗ ···⊗ Vi1 k k Vin .

0.2. Partitions. A partition of an integer n is an ordered increasing sequence (i1, ..., ik) of non-negative integers such that i1 + ···+ ik = n. We denote the set {1, ..., n} by [n]. A partition of the set [n]isaset{I1, ..., Ik} of disjoint subsets of [n] such that I1 ∪···∪Ik =[n].

0.3. Symmetric group and block permutations. We denote the symmetric group of permutations of [n]bySn . Every permutation σ of Sn is written with S ×···×S S S the n-tuple (σ(1), ..., σ(n)). We denote the subgroup i1 in of |¯ı| by ¯ı. From every permutation τ of Sn and every n-tuple ¯ı =(i1, ..., in), one associates a permutation τ¯ı of S|¯ı|, called a block permutation of type ¯ı, deﬁned by

··· −1 ··· −1 τ¯ı = τi1,..., in := (i1 + + iτ (1)−1 +1, ..., i1 + + iτ (1), ...,

i1 + ···+ iτ −1(n)−1 +1, ..., i1 + ···+ iτ −1(n)).

0.4. Gebra. Following the article of J.-P. Serre [S], we call a gebra any algebraic structure like algebras, coalgebras, bialgebras, etc.

1. Composition products In this section we describe an algebraic framework which is used to represent the operations with multiple inputs and multiple outputs acting on algebraic structures. We introduce composition products which correspond to the compositions of such operations.

1.1. S-bimodules. S S P Deﬁnition ( -bimodule). An -bimodule is a collection ( (m, n))m, n∈N,ofk- modules P(m, n) endowed with an action of the symmetric group Sm on the left andanactionofthesymmetricgroupSn on the right such that these two actions are compatible.

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A morphism between two S-bimodules P, Q is a collection of morphisms fm, n : P(m, n) →Q(m, n) which commute with the action of Sm on the left and with the action of Sn on the right. The S-bimodules and their morphisms form a category denoted S-biMod. The S-bimodules are used to code the operations acting on diﬀerent types of gebras. The module P(m, n) represents the operations with n inputs and m outputs: P ⊗ ⊗n → ⊗m (m, n) Sn A A .

Deﬁnition (Reduced S-bimodule). A reduced S-bimodule is an S-bimodule P P P N (m, n) m, n∈N such that (m, 0) = (0,n)=0,foreverym and n in . 1.2. Composition product of S-bimodules. We deﬁne a product in the category of S-bimodules in order to represent to composition of operations.

Deﬁnition (Directed graphs). A directed graph is a non-planar graph where the orientations of the edges are given by a global ﬂow (from the top to the bottom, for instance). We suppose that the inputs and the outputs of each vertex are labelled by integers {1, ..., n}. The global inputs and outputs of each graphs are also supposed to be indexed by integers. We denote the set of such graphs by G.

Every graph studied in this article will be directed.

Deﬁnition (2-level graphs). When the vertices of a directed graph g can be dis- patched on two levels, we say that g is a 2-level graph. In this case, we denote by th 2 Ni the set of vertices on the i level. The set of such graphs is denoted by G (cf. Figure 1)

Figure 1. Example of a 2-level graph.

We can now deﬁne a product in the category of S-bimodules based on 2-level graphs G2. WedenotebyIn(ν)andOut(ν) the sets of inputs and outputs of a vertex ν of a graph.

Deﬁnition (Composition product ). Given two S-bimodules P and Q, we deﬁne their product by the following formula: ⎛ ⎞ ⎝ ⎠ Q P := Q(|Out(ν)|, |In(ν)|) ⊗k P(|Out(ν)|, |In(ν)|) ≈ , 2 g∈G ν∈N2 ν∈N1

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where the equivalence relation ≈ is generated by G >> Ð GG uu >> ÐÐ GG uu >> Ð GG uu >2 ÐÐ σGG(2) uu 1 >ÐÐ 3 σ(1) # zu σ(3) ν ≈ −1 Ñ == τw νσHH 1 ÑÑ ==3 τ(1) w HHτ(3) Ñ 2 = ww HH ÑÑ == ww τ(2) HH ÐÑÑ = ww H$ {w . This product is given by the directed graphs on 2 levels where the vertices are indexed by elements of Q and P with respect to the inputs and outputs. Remark. By concision, we will often omit the equivalence relation in the rest of the paper. It will always be implicit in the following that a graph indexed by elements of an S-bimodule is considered with the equivalence relation. This product has an algebraic writing using symmetric groups. Given two n-tuples ¯ı andj ¯, we use the following conventions. The notation P(¯j, ¯ı) denotes the product P(j1,i1) ⊗k ···⊗k P(jn,in) and the notation S¯ı denotes the S ×···×S S image of the direct product of the groups i1 in in |¯ı|. Theorem 1.1. Let P and Q be two S-bimodules. Their composition product is isomorphic to the S-bimodule given by the formula ⎛ ⎞ ∼ ⎝ ⎠ Q P S ⊗S Q ¯ ¯ ⊗S S ⊗S P ⊗S S ∼ (m, n)= k[ m] l¯ (l, k) k¯ k[ N ] j¯ (¯j, ¯ı) ¯ı k[ n] , N∈N ¯l, k,¯ j,¯ ¯ı where the direct sum runs over the b-tuples ¯l, k¯ and the a-tuples j¯, ¯ı such that |¯l| = m, |k¯| = |j¯| = N, |¯ı| = n and where the equivalence relation ∼ is deﬁned by

θ ⊗ q1 ⊗···⊗qb ⊗ σ ⊗ p1 ⊗···⊗pa ⊗ ω −1 −1 ∼ ⊗ −1 ⊗···⊗ −1 ⊗ ⊗ ⊗···⊗ ⊗ θτ¯l qτ (1) qτ (b) τk¯ σνj¯ pν(1) pν(a) ν¯ı ω, ∈ S ∈ S ∈ S ∈ S for θ m, ω n, σ N and for τ b with τk¯ the corresponding permutation by block ( cf. Conventions), ν ∈ Sa and νj¯ the corresponding permutation by block.

Proof. To any element θ⊗q1⊗···⊗qb⊗σ⊗p1⊗···⊗pa⊗ω of k[Sm]⊗Q(¯l, k¯)⊗k[SN ]⊗ P(¯j, ¯ı) ⊗ k[Sn], we associate a graph Ψ(θ ⊗ q1 ⊗···⊗qb ⊗ σ ⊗ p1 ⊗···⊗pa ⊗ ω)such that its vertices are indexed by the qβ and the pα,for1≤ β ≤ b and 1 ≤ α ≤ a.To do that, we consider a geometric representation of the permutation σ. We gather and index the outputs according to k¯ and the inputs according toj ¯. We index the vertices by the qβ and the pα. Then, for each qβ, we build lβ outgoing edges whose roots are labelled by 1, ..., lβ. We do the same with the pα and the inputs of the graph. Finally, we label the outputs of the graph according to θ and the inputs according to ω. For instance, the element (4123)⊗q1 ⊗q2 ⊗(1324)⊗p1 ⊗p2 ⊗(12435) gives the graph represented in Figure 2. Let Ψ(¯ θ ⊗ q1 ⊗ ··· ⊗ qb ⊗ σ ⊗ p1 ⊗ ··· ⊗ pa ⊗ ω) be the equivalence class of Ψ(θ⊗q1 ⊗···⊗qb ⊗σ⊗p1 ⊗···⊗pa ⊗ω) for the relation ≈. Therefore, the application ¯ S ⊗S Q ¯ ¯ ⊗S S ⊗S Ψ naturally induces an application from the quotient k[ m] l¯ (l, k) k¯ k[ N ] j¯ P ⊗ S ∼ (¯j, ¯ı) S¯ı k[ n]. The equivalence relation corresponds, via Ψ, to a (homeomorphic) rearrangement in space of the graphs. Therefore, the non-planar graph created by Ψ¯ is invariant on the equivalent class of θ ⊗ q1 ⊗···⊗qb ⊗ σ ⊗ p1 ⊗···⊗pa ⊗ ω

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1 @ 2 4 3 @ 5 @@ ~~ @@ ~~ @@ ~~ @@ ~~ @@ ~~ @@ ~~ @2 ~~ @ ~~ _ _1 _ _ 3______1 _ _ 2_ _ p1 PPP nn p2 C 1{{ P2PP n1nn CC2 {{ PPP nnn CC {{ PPPnn CC {{ nn PPP CC {CC {nn CC { CC {{ CC {{ CC {{ CC {{ CC {{ CC {{ C! }{{ C! }{{ _ _1 _ _ 2______1 _ _ 2_ _ q1 q2 @ 1 ~~ @@3 1 ~~ 2 @@ ~~ @@ ~~ @ 4123

Figure 2. Image of (4123) ⊗ q1 ⊗ q2 ⊗ (1324) ⊗ p1 ⊗ p2 ⊗ (12435) under Ψ .

under the relation ∼. This ﬁnally deﬁnes an application Ψ which is an isomorphism of S-bimodules.

Proposition 1.2. The composition product is associative. More precisely, let R, Q and P be three S-bimodules. There is a natural isomorphism of S-bimodules ∼ (R Q) P = R (Q P). Proof. Denote by G3 the set of 3-level graphs. The two S-bimodules (R Q) P and R (Q P) are isomorphic to the S-bimodule given by the following formula: R(|Out(ν)|, |In(ν)|) ⊗ Q(|Out(ν)|, |In(ν)|) ∈G3 ∈N ∈N g ν 3 ν 2 ⊗ P(|Out(ν)|, |In(ν)|) ≈ .

ν∈N1

1.3. Horizontal and connected vertical composition products of S-bimodules. Deﬁnition (Concatenation product ⊗). Let P and Q be two S-bimodules. We deﬁne their concatenation product by the formula P⊗Q S ⊗S ×S P (m, n):= k[ m +m ] m m (m ,n) m+m=m n+n=n ⊗ Q ⊗S ×S S k (m ,n ) n n k[ n +n ]. The product ⊗ corresponds to the intuitive notion of concatenation of operations (cf. Figure 3). The concatenation product ⊗ is also called the horizontal product by contrast with the composition product , which is called the vertical product.

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C 1 :: 2 C { 3 Ô 4 : CC {{{ Ô :: CC ÔÔ :: { C Ô 1 2 }{{ C!1 ÑÔÔ 2 p q Ô ÔÔ 1 1 2ÔÔ ÑÔÔ 13 2

Figure 3. Concatenation of two operations.

Proposition 1.3. The concatenation product induces a symmetric monoidal category structure on the category of S-bimodules. The unit is given by the S-bimodule k deﬁned by k(0, 0) = k, k(m, n)=0 otherwise. Proof. The unit relation comes from P⊗ S ⊗ P ⊗ ⊗ S ∼ P k(m, n)=k[ m] Sm (m, n) k k Sn k[ n] = (m, n), and the associativity relation comes from ∼ P(m, n) ⊗Q(m ,n) ⊗ R(m ,n ) = P(m, n) ⊗ Q(m ,n) ⊗ R(m ,n ) ∼ S ⊗S ×S ×S P ⊗ Q ⊗ = k[ m+m +m ] m m m (m, n) k (m ,n) k R(m ,n ) ⊗S ×S ×S S n n n k[ n+n +n ]. The symmetry isomorphism is given by the following formula: P(m, n) ⊗Q(m ,n) → (1, 2)m, m P(m, n) ⊗Q(m ,n) (1, 2)n, n ∼ = Q(m ,n) ⊗P(m, n).

Remark. The monoidal product ⊗ is bilinear. (It means that the functors P⊗• and •⊗P are additive for every S-bimodule P.) The notions of S-bimodules represents the operations acting on algebraic structures. The composition product models their compositions. For some gebras, one can restrict to connected compositions, based on connected graphs, without loss of information (cf. 2.9). Deﬁnition (Connected graph). A connected graph g is a directed graph which is connected as topological space. We denote the set of such graphs by Gc.

Deﬁnition (Connected composition product c). Given two S-bimodules Q and P, we deﬁne their connected composition product by the following formula: ⎛ ⎞ ⎝ ⎠ Q c P := Q(|Out(ν)|, |In(ν)|) ⊗k P(|Out(ν)|, |In(ν)|) ≈ . ∈G2 ∈N ∈N g c ν 2 ν 1 In this case, the algebraic writing is more complicated. It involves a special class of permutations of the symmetric group SN .

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¯ Deﬁnition (Connected permutations). Let N be an integer. Let k =(k1, ..., kb) be a b-tuple andj ¯ =(j1, ..., ja)beana-tuple such that |k¯| = k1 + ···+ kb = |j¯| = ¯ j1 + ···+ ja = N.A(k, j¯)-connected permutation σ of SN is a permutation of SN such that the graph of a geometric representation of σ is connected if one gathers the inputs labelled by j1 +···+ji +1, ..., j1 +···+ji +ji+1,for0≤ i ≤ a−1, and the outputs labelled by k1 + ···+ ki +1, ..., k1 + ···+ ki + ki+1,for0≤ i ≤ b − 1. ¯ Sc The set of (k, j¯)-connected permutations is denoted by k,¯ j¯ .

Example. Consider the permutation (1324) in S4 and the following geometric representation:

1• 2•6 •3 4• 66 ØØØ 66 ØØ 6 • •Ø • • 1234. Take k¯ =(2, 2) andj ¯ =(2, 2). If one links the inputs 1, 2 and 3, 4 and the outputs 1, 2 and 3, 4, it gives the following connected graph: ? • •?? • • ?? ?? • •• • Therefore, the permutation (1324) is ((2, 2), (2, 2))-connected. ¯ Counterexample. Consider the same permutation (1324) of S4 but let k = (1, 1, 2) andj ¯ =(2, 1, 1). One obtains the following non-connected graph: ? • •?? • • ?? ?? ••• • The next proposition will allow us to link the connected composition product with the one defined in the theory of operads. ¯ Proposition 1.4. If k =(N), all the permutations of SN are ((N), j¯)-connected, Sc S for every j¯. Otherwise stated, one has (N), j¯ = N . ¯ Sc ∅ If k is different from (N), one has k,¯ (1, ..., 1) = . Proof. The proof is obvious. Proposition 1.5. Let P and Q be two S-bimodules. Their connected composition product is isomorphic to the S-bimodule given by the formula ∼ Q c P(m, n) = ⎛ ⎞ ⎝ c ⎠ S ⊗S Q ¯ ¯ ⊗S S ⊗S P ⊗S S ∼ k[ m] l¯ (l, k) k¯ k[ k,¯ j¯] j¯ (¯j, ¯ı) ¯ı k[ n] , N∈N ¯l, k,¯ j,¯ ¯ı where the direct sum runs over the b-tuples ¯l, k¯ and the a-tuples j¯, ¯ı such that |¯l| = m, |k¯| = |j¯| = N, |¯ı| = n. Proof. First, we have to prove that the right member is well defined. Forj ¯a a-tuple ¯ ¯ and k a b-tuple such that |j¯| = |k| = N,wedenote¯j := (jν−1(1), ..., jν−1(b))and ¯ ¯ ¯ k := (kτ −1(1), ..., kτ −1(b)). They also verify |j¯ | = N and |k | = N.Every(k, j¯)- connected permutation σ gives by composition on the left with a block permutation of the type τk¯ and by composition on the right with a bock permutation of the type

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¯ νj¯ a(k , j¯ )-connected permutation. The geometric representations of σ ∈ SN and ◦ ◦ τk¯ σ νj¯ are homeomorphic and one links the same inputs and outputs. Therefore, ∈ Sc ∈ Sc if σ k,¯ j¯, one has τk¯ σνj¯ k¯, j¯ . The rest of the proof uses the same arguments as the proof of Theorem 1.1.

By opposition with the composition product , the connected composition product c is a monoidal product in the category of S-bimodules. It remains to deﬁne the unit in this category. We denote

I(1, 1) = k, I := I(m, n)=0 otherwise.

Proposition 1.6. The category (S-biMod, c,I) is a monoidal category. P Sc Proof. To show the unit relation on c I(m, n), one has to study k,¯ j¯ in the case j¯ =(1, ..., 1). If one does not gather all the outputs, this set is empty. Otherwise, ¯ Sc S for k =(n), one has (n), (1, ..., 1) = n (cf. Proposition 1.4). This gives

⊗n ∼ ⊗n ∼ P P ⊗ S ⊗ ×n P ⊗ ⊗ P c I(m, n)= (m, n) Sn k[ n] S k = (m, n) k.idn k = (m, n). 1 ∼ (The case I c P = P is symmetric.) Once again, the associativity relation comes from the following formula ∼ R c (Q c P) = (R c Q) c P ∼ = R(|Out(ν)|, |In(ν)|) ⊗ Q(|Out(ν)|, |In(ν)|) g∈Gc ν∈N ν∈N 3 3 2 ⊗ P(|Out(ν)|, |In(ν)|) ≈ .

ν∈N1

Examples. The monoidal categories (k-Mod, ⊗k,k)and(S-Mod, ◦,I)aretwofull monoidal subcategories of (S-biMod, c,I).

The ﬁrst category is the category of modules over k endowed with the classical tensor product. Every vector space V can be seen as the following S-bimodule:

V (1, 1) := V, V (j, i):=0 otherwise.

One has V c W (1, 1) = V ⊗k W . Since there exists no connected permutation associated to a, b-tuples of the form (1, ..., 1) (cf. Proposition 1.4), one has V W (j, i)=0for(j, i) =(1 , 1). And the morphisms between two S-bimodules only composed by an (S1, S1)-module are exactly the morphisms of k-modules.

The category of S-bimodules contains the category of S-modules related to the theory of operads (cf. [GK, L3, May]). Given an S-module P, we consider the following S-bimodule:

P(1,n):=P(n)forn ∈ N∗, P(j, i):=0 ifj =1 .

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c We have seen in Proposition 1.4 that S = SN . Here it gives (N), (1, ..., 1)

Q P Q ⊗ S ⊗ ×N P ⊗ S ∼ c (1,n)= (1,N) SN k[ N ] S (1, ¯ı) S¯ı k[ n] , 1 N≤n i1+···+iN =n where the equivalence relation ∼ is given by

q ⊗ σν ⊗ p1 ⊗···⊗pN ∼ q ⊗ σ ⊗ pν(1) ⊗···⊗pν(N).

It is equivalent to take the coinvariants under the action of SN in the expression Q(1,N) ⊗k P(1,i1) ⊗k ···⊗k P(1,iN ). Here we ﬁnd the monoidal product deﬁned in the category of S-modules (cf. [GK, L3, May]), which is well known under the following algebraic form (without the induced representations): QcP(1,n)= Q(1,N) ⊗P(1,i1) ⊗···⊗P(1,iN ) =Q◦P(n).

N≤n i1+···+iN =n SN

Notice that the restriction of the monoidal product c on S-modules corresponds to the description of the composition product ◦ deﬁned by trees (cf. [GK, L3]). When j = 1, the composition Q c P(j, i) represents a concatenation of trees which is a non-connected graph. Therefore, Q c P(j, i)=0,forj =1. The default of the composition product to be a monoidal product, in the category of S-bimodules, comes from the fact that P I corresponds to concatenations of elements of P and not only elements of P.

Deﬁnition (The S-bimodules T⊗(P)andS(P)). Since the monoidal product ⊗ is bilinear, the related free monoid on an S-bimodule P is given by every possible P concatenations of elements of ⊗n T⊗(P):= P , n∈N where P⊗0 = k by convention. The monoidal product is symmetric. Therefore, we can consider the truncated symmetric algebra on P: S P ¯ P P⊗n ( ):=S⊗( )= ( )Sn . n∈N∗

The functor S allows us to link the two composition products and c. Proposition 1.7. Let P and Q be two S-bimodules. One has the following isomorphism of S-bimodules: ∼ S(Q c P) = Q P. Proof. The isomorphism lies on the fact that every graph of G2 is the concatenation G2 of connected graphs of c , and the order between these connected graphs does not matter. Remark that P I = S(P), which is diﬀerent from P, in general. Deﬁnition (Saturated S-bimodules). A saturated S-bimodule P is an S-bimodule such that P = S(P). We denote this category by sat-S-biMod. Example. For every S-bimodule P, S(P) is a saturated S-bimodule. ∼ Notice that S(I)(m, n)=0ifm = n and S(n, n) = k[Sn].

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Proposition 1.8. The category (sat-S-biMod, , S(I)) is a monoidal category. ∼ Proof. Since Q P = S(Q P), the associativity relation comes from Proposi- tion 1.2. If P is a saturated S-bimodule, one has P S(I)=S(P)=P.

We sum up these results in the diagram S (Vect,⊗k,k) → (S-Mod, ◦,I) → (S-biMod, c,I) −→ (sat-S-biMod, , S(I)), where the ﬁrst arrows are faithful functors of monoidal categories. Remark. Since the concatenation product is symmetric and bilinear, its homological properties are well known. In the rest of the text, we will mainly study the homological properties of the composition products (cf. Sections 3 and 5). Since the composition product is obtained by concatenation S from the connected composition product c, we will only state the theorems for the connected composition product c in the following text. There exist analogue theorems for the composition product , which are left to the reader.

1.4. Representation of the elements of QcP. The product QcP (and QP) is isomorphic to a quotient under the equivalence relation ∼ that permutates the order of the elements of Q and P. For instance, to study the homological properties of this product, we need to find natural representatives of the classes of equivalence for the relation ∼. In this section, all the S-bimodules are reduced (cf. 1.1). SB SB Definition ( i, ν ). We denote by i, ν the set of block permutations of type (i,..., i) ν times of Siν (cf. Conventions). SB S Remark. The set i, ν is a subgroup of iν . ∗ Let ¯ı be a partition of the integer n. We consider the n-tuple ¯ı defined by ∗ B n B i := |{j/ij = k}|.DenotebyS the subgroup image of S ∗ in Sn. k ¯ı k=1 k, ik

Proposition 1.9. The product Q c P(m, n) is isomorphic to the S-bimodule de- ﬁned by the following formula: B c B k[S S ] ⊗S Q(¯l, k¯) ⊗S k[S ] ⊗S P(¯j, ¯ı) ⊗S k[S S ], m ¯l l¯ k¯ k,¯ j¯ j¯ ¯ı ¯ı n Θ where the sum Θ runs over ¯ı partition of the integer n, ¯l partition of the integer m, N in N∗ and ja¯ -tuple, kb¯ -tuple such that |k¯| = |j¯| = N. Proof. After choosing the order of the elements of P given by the partition ¯ı of the integer n, the remaining relations that permutate the elements of P involve only SB elements of ¯ı .

We can do the same kind of work with the deﬁnition of c using non-planar graphs. The goal here is to choose a planar representative of indexed graphs. Deﬁnition (Ordered partitions of [n]). An ordered partition of [n] is a sequence (Π1,..., Πk) such that {Π1,..., Πk} is a partition of [n] and such that min(Π1) < min(Π2) < ···

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Let I be a set of i elements. We denote by P(j, i)(I) the module P(j, i) , f :[i]→I where f is a bijection from [i]toI. Every element of P(j, i)(I) can be represented by a pair (x, f). The equivalence relation is deﬁned by (x.τ, f) (x, f ◦ τ) for τ in Si. Therefore, P(j, i)(I)isan(Sj, SI )-module. The module P(j, i)(I) represents the vertices indexed by an element of P(j, i)withi inputs labelled by elements of I.

Proposition 1.10. The product Q c P(m, n) is isomorphic, as an S-bimodule, to c Q | | ⊗ ···⊗ Q | | ⊗S S (Π1) ( Π1 ,k1) k k (Πb) ( Πb ,kb) k¯ k[ k,¯ j¯] Θ ⊗ P | | ⊗ ···⊗ P | | Sj¯ (j1, Π1 )(Π1) k k (ja, Πa )(Πa) , where the sum Θ runs over pairs (Π1,..., Πb), (Π1,..., Πa) of ordered partitions of ([m], [n]), N in N∗ and ja¯ -tuple, kb¯ -tuple such that |k¯| = |j¯| = N.

As a consequence, we will often write the elements of Q c P as Π Π Π Π 1 ⊗···⊗ b ⊗ ⊗ Π1 ⊗···⊗ Πa 1 b Π1 Πa q1 qb σ p1 pa =(q1 , ..., qb ) σ(p1 , ..., pa ), or even without the Π’s.

Remark. The ﬁrst operation p1 has one input indexed by 1, and the operation q1 has one output indexed by 1. We will use this property to build a contracting homotopy for the augmented bar and cobar construction (cf. Section 4.3). The same propositions can be proved for the composition product . One has Sc S to change the set of connected permutations k,¯ j¯ by the symmetric group N .

2. Definitions of properad and prop In this section, we give the deﬁnitions of properad and prop. These notions model the operations acting on gebras. We give the construction of the free properad and prop. Therefore, we can deﬁne the notions of quadratic properad and prop and give examples.

2.1. Definition of properad. Definition (Properad). A properad is a monoid (P,µ,η) in the monoidal category (S-biMod, c,I). Equivalently, one defines a properad by µ • An associative composition P c P −→P, η • and a unit I −→P. Examples. • The inclusions of monoidal categories

(Vect,⊗k,k) → (S-Mod, ◦,I) → (S-biMod, c,I) show that an associative algebra is an operad and that an operad is a properad.

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• The fundamental example of properad is given by the S-bimodule End(V ) of multilinear applications of a vector space V (cf. 2.4). • The main examples of properads studied here are the free and quadratic properads (cf. 2.7, 2.9).

Deﬁnition (Augmented properad). A properad (P,µ,η, ) is called augmented if the counit (or augmentation map) : P→I is a morphism of properads.

Examples. The free properad F(V )(cf. 2.7) and the quadratic properads (cf. 2.9) are augmented properads.

When P is an augmented properad, we have P = I ⊕ P,whereP is the kernel of and is called the ideal of augmentation.

Deﬁnition (Weight graded S-bimodule). A weight graded S-bimodule M is a direct ∈ N S (ρ) sum, indexed by ρ ,of -bimodules M = ρ∈N M .

The morphisms of weight graded S-bimodules are the morphisms of S-bimodules that preserve this decomposition. The set of weight graded S-bimodules with its morphisms forms a category denoted gr-S-biMod.

One can generalize the various products of S-bimodules to the weight graded (ρ) framework. For instance, one has that (Q c P) isgivenbythesumon2-level connected graphs indexed by elements of Q and P such that the total sum of the weight of these elements is equal to ρ.

Deﬁnition (Weight graded properad). A weight graded properad is a monoid in the monoidal category of weight graded S-bimodules with the composition product c.

A weight graded properad P such that P(0) = I is called a connected properad.

2.2. Deﬁnition of prop.

Deﬁnition (Prop). A prop (P, µ,˜ η˜) is a monoid in the monoidal category (sat-S- biMod, , S(I)). In other words, • The S-bimodule P is stable under concatenation P⊗P→P. µ • The composition P P −→P is associative. η • The morphism S(I) −→P is a unit.

Remark. This deﬁnition corresponds to a k-linear version of the notion of PROP (cf. S. MacLane [MacL2] and F.W. Lawvere [La]). A prop is the support of the operations acting on algebraic structures. That’s why we denote it with small letters. The name “properad” is a contraction of “prop” and “operad”.

If one restricts the compositions of a prop to a connected graph, it gives a properad. Therefore, one can consider the forgetful functor Uc from Props to Properads

Uc : Props → Properads.

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Proposition 2.1. The functor S induces a functor from Properads to Props. It is the left adjoint to the forgetful functor Uc:

Uc / Props o Properads. S

Proof. Let (P,µ,η) be a properad. We have seen that the S-bimodule S(P)is saturated. The composition of the prop S(P)isgivenby ∼ S(µ) S(P) S(P) = S(P c P) −−−→S(P), and the unit comes from S(η) S(I) −−−→S(P). The adjunction relation is left to the reader.

2.3. Definition of coproperad. Dually, we define the notion of coproperad. Definition (Coproperad). A coproperad is a comonoid (C, ∆,ε) in the monoidal category (S-biMod, c,I). A structure of coproperad on C is given by ∆ • A coassociative coproduct C −→Cc C, • and a counit C −→ε I. Examples. The main examples of coproperads studied in this article are the cofree connected coproperad F c(V )(cf. 2.8) and the Koszul dual P¡ of a properad P (cf. 7.1.1). 2.4. The example of End(V ). Definition (End(V )). We denote by End(V )theset{linear applications : V ⊗n → ⊗m ⊗n ⊗m V }n, m. The symmetric groups Sn and Sm act on V and V by permutation of the variables. Therefore, End(V )isanS-bimodule. The composi- χ tion End(V ) End(V ) −→ End(V ) is given by the composition of linear applications. And the inclusion η : S(I) → End(V ) corresponds to linear applications V ⊗n → V ⊗n obtained by permutation of the variables.

⊗iα ⊗jα ⊗kβ ⊗lβ For instance, let pα ∈ Homk-Mod(V ,V )andqβ ∈Homk-Mod(V ,V ). ⊗n ⊗m The morphism χ(θ ⊗ q1 ⊗···⊗qb ⊗ σ ⊗ p1 ⊗···⊗pa ⊗ ω) ∈ Homk-Mod(V ,V ) corresponds to the composition ⊗···⊗ ⊗···⊗ ω / p1 pa/ σ / q1 qb / θ / ⊗ V ⊗n V ⊗n V ⊗N V ⊗N V ⊗m V m,

⊗n ⊗i1 ⊗ib ⊗j1 ⊗jb where the morphism p1 ⊗···⊗pa : V = V ⊗···⊗V → V ⊗···⊗V = ⊗N V is the concatenation of morphisms pα.

The S-bimodule End(V ) is saturated and (End(V ),χ,η) is a prop. Its restric- χc tion to connected compositions End(V )c End(V ) −→ End(V ) induces a structure of properad.

2.5. P-module. Here we recall the basic deﬁnitions related to the notion of module over a monoid that will be applied to properads and props throughout the text. For more details, we refer the reader to the book of S. MacLane [MacL1].

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A structure of module (on the right) L over a monoid P in a monoidal category ρ is given by a morphism L c P −→P such that the following diagrams commute:

P P L cµ/ P L cη / P L c c L c L c IK L c KK KK P ρ KK ρ ρ c KK K% ρ / L c P L L. In this case, the object L is called a P-module(ontheright).

The free module on the right on an object M is given by L := M c P.WhenP is a prop, the free P-module on a saturated S-module M is M P.Intherestofthe text, we will call the free P-module on an S-bimodule M, the following P-module S(M) P = M P. Deﬁnition (Relative composition product). Let (L, ρ)bearightP-module and (R, λ)bealeftP-module. Their relative composition product LcP R is given by the cokernel of the following maps:

ρ cR/ L c P c R /L c R. Lcλ

When (P,µ,η, ) is an augmented monoid, one can deﬁne a (left) action of P on I by the formula cI ∼ λ : P c I −−−→ I c I = I. Deﬁnition (Indecomposable quotient). Let (P,µ,η, ) be an augmented monoid in a monoidal category and let (L, ρ)bearightP-module. The indecomposable quotient of L is the relative composition product of L and I, that’s to say, the cokernel of the maps

ρ cI / ∼ L c P c I /L c I = L. Lcλ

When (P,µ,η, ) is an augmented monoid, one has that the indecomposable quotient of the free module L = M c P is isomorphic to M.

2.6. P-gebra. The notion of P-gebra is the natural generalization to the notion of an algebra over an operad. Deﬁnition (P-gebra). Let (P,µ,η) be a properad (or a prop). A structure of P-gebra on a k-module V is given by a morphism of properads (or props) : P→ End(V ). Remark. A P-gebra is an “algebra” over a prop P. The term “algebra” is to be taken here in the largest sense. For instance, a P-gebra can be equipped with coproducts (operations with multiple outputs). It is the case for bialgebras, and Lie bialgebras. Consider the S-module T (V ) deﬁned by T (V )(n):=V ⊗n, where the (left) action is given by the permutation of variables. One can see that a morphism of (left)

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S-modules µV : P V → T (V ) induces a unique morphism of S-modules µV : P T (V ) → T (V ). For instance, the element _ _ _ ⊗ ______⊗ _ _ _ v1 v2QQ lv3 vI4 1 ww QQ2 1 ll II2 ww QQ ll II ww QQQ lll II ww QlQQl II wHw lll QQI I HH vv II uu HH vv II uu HH vv II uu HH vv II uu H# {vv I$ zuu _ _ _1 _ _ 2______1 _ _ 2_ _ _ p1 p2 1 w GG3 1 ww 2 GG ww GG ww GG {ww G# 4 123 can be written ______v1 v3 v2 v4 1 1 1 1 DD z DD z DD zz DD zz DD zz DD zz DD zz DD zz D" |zz D" |zz _ _ 1 _ _ 2______1 _ _ 2 _ _ p1 p2 B 1}} BB3 1 }} 2 BB }} BB }~} B 4123. Proposition 2.2. A P-gebra structure on a k-module V isequivalenttoamor- phism µV : P V → T (V ) such that the following diagram commutes:

P µV / P P V P T (V )

µV µV µV / P V T (V ).

Remark. The same arguments hold for a gebra over a properad. When P is an operad, we ﬁnd the classical notion of an algebra over an operad. 2.7. Free properad and free prop. In this section, we complete the following diagram of forgetful and left adjunct functors:

Uc / Props o Properads O S O F U F U U / sat-S-biMod o S-biMod, S

with the description of the functors F and F . To do this, we will use the construction of the free monoid described in [V1].

We recall the construction of the free properad given in [V1] (section 5).

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Theorem 2.3. The free properad on an S-bimodule V is given by the sum on connected graphs (without level) G with the vertices indexed by elements of V ⎛ ⎞ F(V )=⎝ V (|Out(ν)|, |In(ν)|)⎠ ≈ .

g∈Gc ν∈N

The composition µ comes from the composition of directed graphs.

Remark. If V is an S-module, the connected graphs involved here are trees. There- fore, here we ﬁnd the same construction of the free operad given in [GK]. By the same arguments, we have that the free prop F (V ) on a saturated S- bimodule V is given by the sum on graphs (without level) indexed by elements of V . Here we ﬁnd the same construction of the free prop as B. Enriquez and P. Etingof in [EE].

Deﬁnition (Free prop). In this paper, we call a free prop on an S-module V ,and the image of V by the following composition of left adjunct functors:

F (V ):=S◦F(V ).

Recall from [V1], the notion of split analytic functor.

Deﬁnition (Split analytic functors). A functor f is a split analytic functor if it is a direct sum n∈N f(n) of homogenous polynomial functors of degree n. One can decompose the set of directed (connected) graphs with the number of vertices. We denote by Gc, (n) the set of connected graphs with n vertices. Hence, one has ⎛ ⎞ F(V )=⎝ V (|Out(ν)|, |In(ν)|)⎠ ≈

g∈Gc ν∈N ⎛ ⎞ n ⎝ ⎠ = V (|Out(νi)|, |In(νi)|) ≈ . ∈N ∈G i=1 n g c, (n)

F(n)(V )

Proposition 2.4. The functor F : V →F(V ) is a split analytic functor, where the part of weight n is denoted as F(n)(V ). This graduation is stable under the composition µ of the free properad. Therefore, the free properad is a weight graded properad. We deﬁne the counit by the following projection F(V ) → I = F(0)(V ).With this augmentation map, the free properad is an augmented properad. ∞ S Lemma 2.5. Let f = n=0 f(n) be a split analytic functor in the category of - (ρ) S S bimodules. Let M = ρ∈N M be a weight graded -bimodule. The -bimodule f(M) is bigraded with one graduation induced by the analytic functor and the other one given by the total weight.

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Proof. Denote f(n) = fn ◦ ∆n where fn is an n-linear morphism. The ﬁrst graduation can be written f(M)(n) := fn(M,..., M). The second one corresponds to (ρ) (i1) (in) f(M) := fn(M ,..., M ). n≥1 ··· i1+ +in=ρ

Corollary 2.6. Every free properad on a weight graded S-bimodule is bigraded. 2.8. Cofree connected coproperad. We deﬁne the structure of a cofree connected coproperad on the same S-bimodule as the free properad. Denote ⎛ ⎞ F c(V ):=F(V )=⎝ V (|Out(ν)|, |In(ν)|)⎠ ≈ . g∈Gc ν∈N The projection on the summand generated by the trivial graph gives the counit ε : F c(V ) → I. The coproduct ∆ lies on the set of cuttings of graphs into two parts. The image of an element g(V ) represented by a graph g indexed by operations of V under the coproduct ∆ is given by the formula ∆(g(V )) := g1(V ) c g2(V ),

(g1(V ),g2(V ))

where the sum is over pairs (g1(V ),g2(V )) of family of elements such that

µ(g1(V ) c g2(V )) = g(V ). Proposition 2.7. For every S-bimodule V , (F c(V ), ∆,ε) is a weight graded connected coproperad. This coproperad is cofree in the category of connected coproperads. Proof. The counit relation comes from ⎛ ⎞ ⎝ ⎠ (ε c id) ◦ ∆(g(V )) = (ε c id) ◦ ∆ g1(V ) c g2(V ) (g (V ),g (V )) 1 2 = ε(g1(V )) c g2(V )

(g1(V ),g2(V ))

= I c g(V ) = g(V ). The coassociativity relation comes from (∆ id) ◦ ∆(g(V )) = (id ∆) ◦ ∆(g(V )) c c = g1(V ) c g2(V ) c g3(V ),

(g1(V ),g2(V ),g3(V ))

where the sum is over the triples (g1(V ),g2(V ),g3(V )) such that

µ(g1(V ) c g2(V ) c g3(V )) = g(V ). Let C be a connected coproperad and let f : C → V be a morphism of S- bimodules. The analytic decomposition of F c(V ), according to the number of vertices of graphs, and the fact that C is connected allows us to deﬁne by induction a

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morphism of connected coproperads f¯ : C →Fc(V ). This morphism is determined by f and is the unique morphism of connected coproperads such that the following diagram commutes: o V bE F c(V ) EE O EE EE f¯ f EE C.

Remark. This construction generalizes the construction given by T. Fox in [F] for cofree connected coalgebras. 2.9. Quadratic properads and quadratic props. We deﬁne the notions of quadratic properad and quadratic prop. We relate these two notions and give examples.

In this paper, we are interested in the study of properads and props defined by generators and relations. Let V be an S-bimodule and R be a sub-S-bimodule of F(V ). Consider the ideal of F(V ) generated by R. (For the notion of ideal in a monoidal category, we refer the reader to [V2].) The quotient properad F(V )/(R) is generated by V and the relations R. The Koszul duality for props deals with a subclass of properads and props composed by quadratic properads and quadratic props. Definition (Quadratic properads). A quadratic properad is a properad defined by generators V and relations R such that R is a sub-S-bimodule of F(2)(V ) (the part of weight 2 of F(V )). It means that R is composed by linear combinations of connected graphs indexed by 2 elements of V . Proposition 2.8. Every properad P = F(V )/(R) defined by generators V and relations R such that R is a sub-S-bimodule of F(n)(V ) is a weight graded properad. Proof. Since the ideal (R) is generated by a homogenous weight graded S-bimodule, the quotient properad F(V )/(R) is weight graded. The elements of weight n of P correspond to the image of F(n)(V ) under the projection F(V ) P = F(V )/(R).

Corollary 2.9. Every quadratic properad is weight graded. Once again, the same results hold for props. Deﬁnition (Quadratic props). A quadratic prop is a prop deﬁned by a generating

S-bimodule V and relations R such that R is a sub-S-bimodule of F(2)(V ) (the part of weight 2 of F (V )). Quadratic properads and quadratic props are related by the following proposition.

Proposition 2.10. Let V be an S-bimodule and R asub-S-bimodule of F(2)(V ) ⊂

F(2)(V ). The quotient prop F(V )/(R) is isomorphic to S(F(V )/(R)),where F(V )/(R) is the quadratic properad generated by V and R.

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Proof. Consider the morphism of props

F (V )=F◦S (V )=S◦F(V ) −→SΦ (F(V )/(R)). Since its kernel is the ideal generated by R in F(V ), Φ induces an isomorphism ¯ F (V )/(R) −→SΦ (F(V )/(R)).

Remark. This proposition implies that, when the relations of a quadratic prop are deﬁned by linear combinations of connected graphs, it is enough to study the related quadratic properad. In this case, the quadratic prop is obtained by concatenation from the quadratic properad. Since the concatenation product is well known in homology, it is enough to study the properad associated to some algebraic structures (gebras) to understand their homological properties (deformation, gebra up to homotopy) (cf. 7.4). Examples. Once again, the ﬁrst examples come from the full subcategories k- Mod and S-Mod. Quadratic algebras, like S(V )andΛ(V ) (the original examples of J. - L. Koszul), are quadratic properads. Also, quadratic operads, like As, Com and Lie (coding associative, commutative and Lie algebras), are quadratic properads. Examples. Let us give new examples treated by this theory. All the following properads are quadratic: • The properad BiLie coding Lie bialgebras. This properad is generated by the following S-bimodule: ⎧ ⎨ ⊗ 12 V (1, 2) = λ.k sgnS2 , ? ?? ? V = V (2, 1) = ∆.sgnS ⊗ k, = ⊗ sgnS ⊕ ⊗ sgnS , ⎩ 2 2 2 V (m, n)=0 otherwise, 12 S where sgnS2 is the signature representation of 2. Its quadratic relations are given by ⎧ ⎪ ⎨⎪ λ(λ, 1) (123) + (231) + (312) ⊕ (123) + (231) + (312) (∆, 1)∆ R = ⎪ ⊕ ⊗ − ⊗ ⊗ ⎩⎪ ∆ λ (λ, 1) (213) (1, ∆) −(1,λ)(∆, 1) − (1,λ)(∆, 1) − (1,λ) ⊗ (132) ⊗ (∆, 1) ⎧ ⎪ 123? 231? 312? ⎪ ?? ?? ?? ⎪ ? + ? + ? ⎪ ⎪ ⎪ ⎪ ? ? ? ⎨⎪ ??? ??? ??? ⊕ ? ? + ? ? + ? ? = ⎪ ⎪ 123 231 312 ⎪ ⎪ 12 12 21 12 21 ⎪ ?? ⎪ ?? ?? ?? ?? ?? ?? ⎪ ⊕ ?? − + − + . ⎩⎪ 12 12 12 12 12 The BiLie-gebras are exactly the Lie bialgebras introduced by V. Drin- feld (cf. [Dr1]). • One variation of Lie bialgebras has been introduced by M. Chas (cf. [Chas]) in the framework of String Topology. An involutive Lie bialgebra is a Lie bialgebra such that λ ◦ ∆ = 0. Therefore, the associated properad is the

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?? same than BiLie,wherethe“loop”?? is added to the space of relations. We denote it by BiLie0, since every composition based on level graphs of genus > 0 is null. • The properad εBi coding inﬁnitesimal Hopf algebras. This properad is generated by the S-bimodule ⎧ ⎨ ⊗ S ? V (1, 2) = m.k k[ 2], ? ? S ⊗ ⊕ ? V = ⎩ V (2, 1) = ∆.k[ 2] k, = , V (m, n)=0 otherwise.

Its relations are given by ⎧ ⎨ m(m, 1) − m(1,m) ⊕ − R = ⎩ (∆, 1)∆ (1, ∆)∆ ⊕ ∆ ⊗ m − (m, 1)(1, ∆) − (1,m)(∆, 1) ⎧ ? ? ? ⎪ ?? ?? ? ⎪ ? − ? ⎪ ⎨⎪ ? ? ⊕ ??? − ?? = ⎪ ? ? ? ⎪ ⎪ ? ⎪ ? ? ? ? ⎩ ⊕ ? − ?− ?? ? .

These εBi-gebras are associative and non-commutative analogs of Lie bialgebras. They correspond to the inﬁnitesimal Hopf algebras deﬁned by M. Aguiar (cf. [Ag1], [Ag2] and [Ag3]). • 1 B The properad 2 i, the degenerated analogue of the properad of bialgebras. The generating S-bimodule V is the same as in the case of εBi,composed by a product and a coproduct. The relations R are the following ones: ⎧ ⎨ m(m, 1) − m(1,m) ⊕ − R = ⎩ (∆, 1)∆ (1, ∆)∆ ⊕ ∆ ⊗ m (κ) ⎧ ? ? ? ⎪ ?? ?? ? ⎪ ? − ? ⎪ ⎨⎪ ? ? ⊕ ??? − ?? = ⎪ ? ? ? ⎪ ⎪ ? ⎪ ? ⎩ ⊕ ? ? (κ).

This example has been introduced by M. Markl in [Ma2] to describe a minimal model for the prop of bialgebras.

Counterexamples. • The previous properad is a quadratic version of the properad Bi coding bialgebras. This last one is deﬁned by generators and relations but is not B 1 B quadratic. The deﬁnition of i is the same as 2 i where the relation (κ)

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is replaced by the following one:

(κ):∆⊗ m − (m, m) ⊗ (1324) ⊗ (∆, ∆) ? ? wG wKKK ? − GwG wwGG ss = ? w s .

Since the relation (κ) includes a composition involving 4 generating operations, the properad Bi is neither quadratic nor weight graded. • Consider the prop coding unitary infinitesimal Hopf algebras defined by J.-L. Loday in [L4]. Its definition is the same as the prop εBi where the third relation is ? ? ? ? ? ⊗ − − − ? − ?− ??− ∆ m (m, 1)(1, ∆) (1,m)(∆, 1) (1, 1) = ? . Since this relation is neither connected nor homogenous of weight 2, one has that this prop is not quadratic and it does not come from a properad.

3. Differential graded framework In this section, we generalize the previous notions in the differential graded framework. For instance, we define the compositions products of a differential graded S-bimodule. We give the definitions of quasi-free P-modules and quasi-free properads.

3.1. The category of differential graded S-bimodules. We now work in the category dg-Mod of differential graded k-modules. Recall that this category can be endowed with a tensor product V ⊗k W defined by the following formula: (V ⊗k W )d := Vi ⊗k Wj, i+j=d

where the boundary map on V ⊗k W is given by

δ(v ⊗ w):=δ(v) ⊗ w +(−1)|v|v ⊗ δ(w),

for v ⊗ w an elementary tensor and where |v| denotes the homological degree of v. The symmetry isomorphisms involve Koszul-Quillen sign rules

|v||w| τv, w : v ⊗ w → (−1) w ⊗ v.

More generally, the commutation of two elements (morphisms, diﬀerentials, objects, etc.) of degree d and e induces a sign (−1)de.

Deﬁnition (dg-S-bimodule). A dg-S-bimodule P is a collection (P(m, n))m, n∈N of diﬀerential graded modules with an action of the symmetric group Sm on the left and an action of Sn on the right. These groups acts by morphisms of dg-modules. We denote by Pd(m, n) the sub-(Sm, Sn)-module composed by elements of degree d of the chain complex P(m, n).

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3.2. Composition products of dg-S-bimodules. The composition products c and of S-bimodules can be generalized to the diﬀerential graded framework. The deﬁnition remains the same except that the equivalence relation ∼ is based on symmetry isomorphisms and therefore involves Koszul-Quillen sign rules. For instance, one has

θ ⊗ q1 ⊗···⊗qi ⊗ qi+1 ⊗···⊗qb ⊗ σ ⊗ p1 ⊗···⊗pa ⊗ ω

|qi||qi+1| ∼ (−1) θ ⊗ q1 ⊗···⊗qi+1 ⊗ qi ⊗···⊗qb ⊗ σ ⊗ p1 ⊗···⊗pa ⊗ ω, −1 where σ =(1...i+1i... a)k¯ σ and θ = θ(1 ...i+1i... a)¯l . Remark. To lighten the notations, we will often omit the induced representations θ and ω in the following text.

⊗···⊗ ⊗ ⊗ ⊗···⊗ Q ¯ ¯ ⊗S We deﬁne the image of an element q1 qb σ p1 pa of (l, k) k¯ c k[S ] ⊗S P(¯j, ¯ı) under the boundary map δ by the following formula: k,¯ j¯ l¯

δ(q1 ⊗···⊗qb ⊗ σ ⊗ p1 ⊗···⊗pa) b |q1|+···+|qβ−1| = (−1) q1 ⊗···⊗δ(qβ) ⊗···⊗qb ⊗ σ ⊗ p1 ⊗···⊗pa β=1 a |q1|+···+|qb|+|p1|+···+|pα−1| + (−1) q1 ⊗···⊗qb ⊗ σ ⊗p1 ⊗···⊗δ(pα) ⊗···⊗pa. α=1 Lemma 3.1. The differential δ is constant on equivalence classes under the relation ∼. Proof. The proof is straightforward and left to the reader. One can also generalize the concatenation product ⊗ to the category of dg-S- bimodules. Definition (Saturated differential graded S-bimodule). A saturated differential ∼ graded S-bimodule is a dg-S-bimodule P such that S(P) = P.Wedenotethis category sat-dg-S-biMod. One can extend Proposition 1.9 and Proposition 1.10 in the differential graded framework. Proposition 3.2. Let Q and P be two differential graded S-bimodules. Their connected composition product Q c P(m, n) is isomorphic to the dg-S-bimodule B c B k[S S ] ⊗S Q(¯l, k¯) ⊗S k[S ] ⊗S P(¯j, ¯ı) ⊗S k[S S ] m ¯l l¯ k¯ k,¯ j¯ j¯ ¯ı ¯ı n Θ and to the dg-S-bimodule c (Π )Q(|Π |,k) ⊗ ···⊗ (Π )Q(|Π |,k) ⊗S k[S ] 1 1 1 k k b b b k¯ k,¯ j¯ Θ ⊗ P | | ⊗ ···⊗ P | | Sj¯ (j1, Π1 )(Π1) k k (ja, Πa )(Πa) . Once again, we have the same propositions for the composition product .We are going to study the homological properties of the composition product of two dg-S-bimodules.

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Remark that the chain complex (Q c P,δ) corresponds to the total complex of a bicomplex. We deﬁne the bidegree of an element B (q , ..., q ) σ (p ,..., p )ofk[S S ] ⊗S Q(¯l, k¯) 1 b 1 a m ¯l l¯ c B ⊗S k[S ] ⊗S P(¯j, ¯ı) ⊗S k[S S ] k¯ k,¯ j¯ j¯ ¯ı ¯ı n

by (|q1|+···+|qb|, |p1|+···+|pa|). The horizontal diﬀerential δh : Qc P→Qc P is induced by the one of Q and the vertical diﬀerential δv : Q c P→Qc P is induced by the one of P. One has explicitly, δh (q1, ..., qb) σ (p1,..., pa) b |q1|+···+|qβ−1| = (−1) (q1,..., δ(qβ),..., qb) σ (p1,..., pa) β=1 and δv (q1, ..., qb) σ (p1,..., pa) a |q1|+···+|qb|+|p1|+···+|pα−1| = (−1) (q1,..., qb) σ (p1,..., δ(pα),..., pa). α=1

Onecanseethatδ = δh + δv and that δh ◦ δv = −δv ◦ δh. Like every bicomplex, this one gives rise to two spectral sequences Ir(Q P) r and II (Q P) that converge to the total homology H∗(Q P,δ). Recall that these two spectral sequences verify

1 2 I (Q c P)=H∗(Q c P,δ),I(Q c P)=H∗(H∗(Q c P,δv),δh) 1 2 and II (Q c P)=H∗(Q c P,δh),II(Q c P)=H∗(H∗(Q c P,δh),δv).

Proposition 3.2 shows that the composition product Q c P can be decomposed as the direct sum of chain complexes B c B S S ⊗S Q ¯ ¯ ⊗S S ⊗S P ⊗S S S k[ m ¯l ] l¯ (l, k) k¯ k[ k,¯ j¯] j¯ (¯j, ¯ı) ¯ı k[ ¯ı n]. Θ

This decomposition is compatible with the structure of bicomplex. Therefore, the spectral sequences can be decomposed in the same way r r B I (Q P)= I k[S S ] ⊗S Q(¯l, k¯) c m ¯l l¯ Θ c B ⊗S S ⊗S P S S k¯ k[ k,¯ j¯] j¯ (¯j, ¯ı)k[ ¯ı n] r r B c and II (Q P)= II k[S S ] ⊗S Q(¯l, k¯) ⊗S k[S ] c m ¯l l¯ k¯ k,¯ j¯ Θ ⊗ P ⊗ SB S Sj¯ (¯j, ¯ı) S¯ı k[ ¯ı n]

From these two formulas, one has the following proposition.

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Proposition 3.3. When k is a ﬁeld of characteristic 0, one has 1 B c B I k[S S ] ⊗S Q(¯l, k¯) ⊗S k[S ] ⊗S P(¯j, ¯ı) ⊗S k[S S ] m ¯l l¯ k¯ k,¯ j¯ j¯ ¯ı ¯ı n B c B S S ⊗S Q ¯ ¯ ⊗S S ⊗S ∗ P ⊗S S S = k[ m ¯l ] l¯ (l, k) k¯ k[ k,¯ j¯] j¯ H ( (¯j, ¯ı)) ¯ı k[ ¯ı n] and 1 B c B II k[S S ] ⊗S Q(¯l, k¯) ⊗S k[S ] ⊗S P(¯j, ¯ı) ⊗S k[S S ] m ¯l l¯ k¯ k,¯ j¯ j¯ ¯ı ¯ı n B c B = k[S S ] ⊗S H∗ Q(¯l, k¯) ⊗S k[S ] ⊗S P(¯j, ¯ı) ⊗S k[S S ]. m ¯l l¯ k¯ k,¯ j¯ j¯ ¯ı ¯ı n S S Proof. By Mashke’s Theorem, we have that the rings of the form k[ j¯]andk[ ¯ı] B c are semi-simple. Therefore, the modules k[S S ] ⊗S Q(¯l, k¯) ⊗S k[S ]arek[S ] m ¯l l¯ k¯ k,¯ j¯ j¯ SB S S projective modules (on the right) and the modules k[ ¯ı n]arek[ ¯ı] projective modules (on the left) .

Proposition 3.4. When k is a ﬁeld of characteristic 0, one has 2 B c B I k[S S ] ⊗S Q(¯l, k¯) ⊗S k[S ] ⊗S P(¯j, ¯ı) ⊗S k[S S ] m ¯l l¯ k¯ k,¯ j¯ j¯ ¯ı ¯ı n 2 B c B S S ⊗S Q ¯ ¯ ⊗S S ⊗S P ⊗S S S = II k[ m ¯l ] l¯ (l, k) k¯ k[ k,¯ j¯] j¯ (¯j, ¯ı) ¯ı k[ ¯ı n] B c B S S ⊗S ∗Q ¯ ¯ ⊗S S ⊗S ∗P ⊗S S S = k[ m ¯l ] l¯ H (l, k) k¯ k[ k,¯ j¯] j¯ H (¯j, ¯ı) ¯ı k[ ¯ı n]. Proof. We use the same arguments to show that 2 B c B S S ⊗S Q ¯ ¯ ⊗S S ⊗S P ⊗S S S I k[ m ¯l ] l¯ (l, k) k¯ k[ k,¯ j¯] j¯ (¯j, ¯ı) ¯ı k[ ¯ı n] B c B S S ⊗S ∗ Q ¯ ¯ ⊗S S ⊗S ∗ P ⊗S S S = k[ m ¯l ] l¯ H (l, k) k¯ k[ k,¯ j¯] j¯ H ( (¯j, ¯ı)) ¯ı k[ ¯ı n]. We conclude by K¨unneth’s formula ¯ ¯ H∗ Q(l, k) = H∗ (Q(l1,k1) ⊗···⊗Q(la,ka)) ∼ ¯ ¯ = H∗Q(l1,k1) ⊗···⊗H∗Q(la,ka)=H∗Q(l, k).

Remark. Let Φ : M → M and Ψ : N → N be two morphisms of dg-S-bimodules. The morphism Φ c Ψ:M c N → M c N is a morphism of bicomplexes. It r r r induces morphisms of spectral sequences I (Φc Ψ) : I (M c N) → I (M c N ) r r r and II (Φ c Ψ) : II (M c N) → II (M c N ). More generally, we have the following proposition. Proposition 3.5. When k is a ﬁeld of characteristic 0, one has

H∗(Q c P)(m, n) B c B = k[S S ] ⊗S H∗Q(¯l, k¯) ⊗S k[S ] ⊗S H∗P(¯j, ¯ı) ⊗S k[S S ]. m ¯l l¯ k¯ k,¯ j¯ j¯ ¯ı ¯ı n Θ

That’s to say, H∗(Q c P)=(H∗Q) c (H∗P). Proof. It is a direct corollary of Mashke’s Theorem (for coinvariants) and K¨unneth’s Theorem (for tensor products).

Once again, similar results apply to the connected composition products .

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Remark. These propositions generalize the results of B. Fresse for the composition product ◦ of operads to the composition products c and of properads and props (cf. [Fr] 2.3). 3.3. Differential properads and differential props. We give here a differential variation of the notions of properads and props. Definition (Differential properad). A differential properad (P,µ,η)isamonoid in the monoidal category (dg-S-biMod, c,I) of differential graded S-bimodules. The composition morphism µ of a differential properad is a morphism of chain complexes of degree 0 compatible with the action of the symmetric groups Sm and Sn. One has the derivation type relation: δ (µ((p1,..., pb) σ (p1,..., pa))) b | | ··· | | − p1 + + pβ−1 = ( 1) µ((p1,..., δ(pβ),..., pb) σ (p1,..., pa)) β=1 a | | ··· | | | | ··· | | − p1 + + pb + p1 + + pα−1 + ( 1) µ((p1,..., pb) σ (p1,..., δ(pα),..., pa)). α=1 Proposition 3.5 gives the following property. Proposition 3.6. When k is a field of characteristic 0, the homology groups (H∗(P),H∗(µ),H∗(η)) of a differential properad (P,µ,η) form a graded properad.

Definition (Differential weight graded S-bimodule). A differential weight graded S- ∈ N S (ρ) bimodule M is a direct sum over ρ of differential -bimodules M = ρ∈N M . The morphisms of differential weight graded S-bimodules are the morphisms of differential S-bimodules that preserve this decomposition. This forms a category denoted gr-dg-S-biMod. Remark that the weight is a graduation separate from the homological degree that gives extra information. Definition (Differential weight graded properad). A differential weight graded properad is a monoid in the monoidal category of differential weight graded S- bimodules. A differential weight graded properad P such that P(0) = I is called connected. Remark. Dually, we can define the notion of differential coproperad and weight graded differential coproperad. A differential coproperad is a coproperad where the coproduct is a morphism of dg-S-bimodules verifying a coderivation type relation. Similarly one has the notions of differential weight graded prop and coprop. 3.4. Free and quasi-free P-modules. We define a variation of the notion of a free P-module, namely the notion of quasi-free P-module, and we prove some of its basic properties. Let P be a (weight graded) dg-properad. Recall from 2.5, that the free differential P-module on the right over a dg-S-bimodule M is given by L = M c P,wherethe differential δ comes from the one of M and the one of P as explained before. When (P,µ,η, ) is an augmented dg-properad, one has that the indecomposable quotient ofthefreedg-P-module L = M c P is isomorphic to M,asadg-S-bimodule.

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In the following part of the text, we will encounter differential P-modules which are not strictly free. They are built on the dg-S-bimodule M c P, but the boundary map is the sum of the canonical differential δ with an homogenous morphism of degree −1. Definition (Quasi-free P-module). Let P be a differential properad. A (right) quasi-free P-module L is a dg-S-bimodule of the form M c P, where the differential δθ is the sum δ + dθ of the canonical differential δ on the product M c P with a homogenous morphism dθ : M c P→M c P of degree −1 such that

dθ ((m1,..., mb) σ (p1,..., pa)) b |m1|+···+|mβ−1| = (−1) (m1,..., dθ(mβ),..., mb) σ (p1,..., pa). β=1

In the weight graded framework, we ask dθ to be a homogenous map for the weight such that (ρ) → P (ρ) dθ : M M c . (<ρ) (>1)

In this case, the boundary map δθ preserves the global weight of L.

Therefore, the boundary map δθ : M c P→M c P veriﬁes the following relation:

δθ ((m1,..., mb) σ (p1,..., pa)) b |m1|+···+|mβ−1| = (−1) (m1,..., δθ(mβ),..., mb) σ (p1,..., pa) β=1 a |m1|+···+|mb|+|p1|+···+|pα−1| + (−1) (m1,..., mb) σ (p1,..., δ(pα),..., pa). α=1 2 2 2 Since δ =0,theidentityδθ = 0 is equivalent to δdθ + dθδ + dθ =0. ∈ P If one writes dθ(mβ)=(m1,..., mb ) σ (p1,..., pa ) M c , then one has

(m1,..., dθ(mβ),..., mb) σ (p1,..., pa) × ∈ P (m1,..., m1,..., mb ,..., mb) σ µ ((p1,..., pa ) σ (p1,..., pa)) M c . M cη/ Notice that the morphism dθ is determined by its restriction on M M c P .

M cη/ Proposition 3.7. The morphism M M c P is a morphism of dg-S-bimodules if and only if dθ =0.

When P is an augmented dg-properad, the condition for the morphism M c ε to be a morphism of diﬀerential graded modules is weaker. Proposition 3.8. Let (P,µ,η,ε) be an augmented dg-properad. The morphism M cε / M c P M is a morphism of dg-S-bimodules if and only if the following composition is null:

dθ / M cε / M c P M c P M =0.

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Corollary 3.9. In this case, the indecomposable quotient of L = M c P is isomorphic to M as a dg-S-bimodule.

Deﬁnition (Morphism of weight graded quasi-free P-module). Let L = McP and L = M c P be two weight graded quasi-free P-modules. A morphism Φ : L → L of weight graded diﬀerential P-modules is a morphism of weight graded quasi-free P-modules if Φ preserves the graduation on the left P→ P Φ:M c M c . (ρ) (ρ)

In the following of the text, the examples of weight graded quasi-free P-modules that we will have to treat will have a particular form.

Definition (Analytic quasi-free P-module). Let P be a weight graded differential properad. An analytic quasi-free P-module is a quasi-free P-module L = M c P such that the weight graded differential S-bimodule M is given by the image M = Υ(V ) of a weight graded differential S-bimodule V under a split analytic functor Υ (cf. 2.7). Denote Υ = n∈N Υ(n),whereΥ(n) is a homogenous polynomial functor (0) of degree n and such that Υ(0) = I.Moreover,V must verify V =0sothat (0) Υ(V ) = I.AswehaveseeninLemma2.5,Υ(V ) is bigraded. The morphism dθ is supposed to verify

(ρ) → P dθ :Υ(n)(V ) Υ(V ) c , n−1 1

where the graduation (ρ) is the global graduation coming from the one of V and (n) is the degree of the analytic functor Υ.

Examples. The examples of analytic quasi-free P-modules fall into two parts: • When V = P, we have the example of the (diﬀerential) bar construction Υ(V )=B¯(P)=F c(ΣP)(cf. Section 4). The augmented bar construction B¯(P) c P is an analytic quasi-free P-modules. • When P is quadratic properad generated by a dg-S-bimodule V , the Koszul dual coproperad P¡ is a split analytic functor in V (cf. Section 7). There- ¡ fore, the Koszul complex P c P is an analytic quasi-free P-module.

We have immediately the following proposition.

Proposition 3.10. Let P be an augmented weight graded dg-properad and L = Υ(V ) c P be an analytic quasi-free P-module. We have

dθ / M cε / M c P M c P M =0.

Therefore, we can identify the indecomposable quotient of L to Υ(V ).

Remark. Dually, we have the notions of a diﬀerential cofree C-comodule on a dg- S-bimodule M,givenbyM c C, a quasi-cofree C-comodule and an analytic quasi- c cofree C-comodule. The augmented cobar construction B¯ (C) c C is an example of an analytic quasi-free C-comodule.

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3.5. Free and quasi-free properads. Here we define the notion of a free differential properad. As in the case of P-modules, we will have to deal with differential properads that are not strictly free. Therefore, we introduce the notion of a quasi- free properad which is a free properad where the differential is given by the sum of the canonical one with a derivation. Let (V, δ)beadg-S-bimodule. We have seen in 2.7 that the free properad on V is given by the weight graded S-bimodule F(V ) which is the direct sum on connected graphs (without level) where the vertices are indexed by elements of V , ⎛ ⎞ F(V )=⎝ V (|Out(ν)|, |In(ν)|)⎠ ≈ .

g∈Gc ν∈N This construction corresponds to the quotient of the simplicial S-bimodule FS cn ≡ (V )= n∈N(V+) under the relation generated by I c V V c I.In the differential graded framework, one has to take the signs into account. In this case, we quotient FS(V )bytherelation | | | | ≡ − ν2 + ν1 (ν1,ν,ν2) σ (ν1, 1,ν2) ( 1) (ν1, 1, 1,ν2) σ (ν1,ν,ν2), for instance, where ν belongs to V (2, 1). Proposition 3.11. The canonical differential δ defined on FS(V ) by the one of V is constant on the equivalence classes for the relation ≡. Proof. The verification of signs is straightforward and is the same as in Lemma 3.1.

Theorem 3.12. The differential properad (F(V ),δ) is free on the dg-S-bimodule (V, δ). Remark. Since the analytic decomposition of the functor F(V ) given in Proposi- tion 2.4 is stable under the differential δ,wehavethatF(V ) is a weight graded differential properad. The projection F(V ) → I is a morphism of dg-properads. The functor F shares interesting homological properties. Proposition 3.13. Over a field k of characteristic 0,thefunctorF : dg-S-biMod → dg-properad is exact. We have H∗(F(V )) = F(H∗(V )). Proof. The proof of this proposition is the same as the proof of Theorem 4 of the article of M. Markl and A. A. Voronov [MV] and lies, once again, on Mashke’s theorem. Definition (Derivation). Let P be a dg-properad. A homogenous morphism of S- bimodules d : P→Pof (homological) degree |d| is called a derivation if it satisfies the following identity: δ (µ((p1,..., pb) σ (p1,..., pa))) b | | ··· | | | | − ( p1 + + pβ−1 ) d = ( 1) µ((p1,..., δ(pβ),..., pb) σ (p1,..., pa)) β=1 a | | ··· | | | | ··· | | | | − ( p1 + + pb + p1 + + pα−1 ) d + ( 1) µ((p1,..., pb) σ (p1,..., δ(pα),..., pa)). α=1

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For example, the differential δ of a differential properad is a derivation of degree −1 such that δ2 = 0. Notice that the sum δ+d of the differential δ with a derivation d of degree −1 is a derivation of the same degree. In this case, since δ2 =0the equation (δ+d)2 is equivalent to δd + dδ + d2 =0. Remark. Dually, one can define the notion of coderivation on a differential coproperad C. The next proposition gives the form of the derivations on the free differential properad F(V ). Proposition 3.14. Let F(V ) be the free dg-properad on the dg-S-bimodule V . For any homogenous morphism θ : V →F(V ), there exists a unique derivation dθ : F(V ) →F(V ) with the same degree such that its restriction to V is equal to θ. This correspondence is one-to-one. Moreover, if θ : V →F(r)(V ) ⊂F(V ), we have dθ F(s)(V ) ⊂F(r+s−1)(V ).

Proof. Let g be a graph with n vertices. We deﬁne dθ on a representative of an equivalence class for the relation ≡ (vertical move of a vertex up to sign) associated to the graph g. This corresponds to a choice of an order for! writing the elements n ∈ of V that index the vertices of g. Write this representative i=1 νi,withνi V . Hence, we deﬁne d by the following formula: θ ⎛ ⎞ ⎛ ⎞ n n i−1 n |ν1|+···+|νi−1| ⎝ ⎠ ⎝ ⎠ dθ( νi)= (−1) νj ⊗ θ(νi) ⊗ νj . i=1 i=1 j=1 j=i+1

Onecanseethatdθ is well deﬁned on the equivalence classes for the relation ≡. Applying dθ is given by applying θ to every element of V indexing a vertex of g. (Since the application θ is an application of S-bimodules, the morphism dθ is well deﬁned on the equivalence classes for the relation ≈ .) The product µ of the free properad F(V ) is surjective. Hence, it shows that every derivation is of this form. If the image under θ of the elements of V is a sum of elements of F(V )thatcan be written by graphs with r vertices (θ creates r − 1 vertices), the form of dθ shows that dθ F(s)(V ) ⊂F(r+s−1)(V ). Dually, we have the same proposition for the coderivations of the cofree connected coproperad. Proposition 3.15. Let F c(V ) be the cofree connected dg-coproperad on the dg-S- bimodule V . For any homogenous morphism θ : F c(V ) → V , there exists a unique c c coderivation dθ : F (V ) →F (V ) such that the composition

dθ / proj / F c(V ) F c(V ) V is equal to θ. This correspondence is one-to-one. F c → F c ⊂Fc Moreover, if θ : (V ) V is null on the components (r)(V ) (V ),for s = r, we have dθ F(s+r−1)(V ) ⊂F(s)(V ), for all s>0. c Graphically, applying the coderivation dθ on an element of F (V ) represented by a graph g corresponds to applying θ on every admissible sub-graph of g. We can now state the deﬁnition of a quasi-free properad.

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Definition. (Quasi-free properad) A quasi-free properad (P,δθ) is a dg-properad of the form P = F(V ), with (V, δ)adg-S-bimodule and where the differential δθ : F(V ) →F(V ) is the sum of the canonical differential δ with a derivation dθ of degree −1, δθ = δ + dθ.

The previous proposition shows that every derivation dθ on F(V ) is determined by its restriction θ on V . Therefore, the inclusion V →F(V ) is a morphism of dg-S-bimodules if and only if θ is null. On the contrary, the projection F(V ) → V is a morphism of dg-S-bimodules under a weaker condition.

Proposition 3.16.The projection F(V ) → V is a morphism of dg-S-bimodules if ⊂ F and only if θ(V ) r≥2 (r)(V ). In this case, we say that the diﬀerential is decomposable. Dually, we have the notion of a quasi-cofree coproperad.

Definition (Quasi-cofree coproperad). A quasi-cofree coproperad (C,δθ)isadg- coproperad of the form C = F c(V ), with (V, δ)adg-S-bimodule and where the c c differential δθ : F (V ) →F (V ) is the sum of the canonical differential δ with a coderivation dθ of degree −1, δθ = δ + dθ. The previous proposition on coderivations of F c(V ) shows that the projection F c(V ) → V is a morphism of dg-S-bimodules if and only if θ is null and that the inclusion V →Fc(V ) is a morphism of dg-S-bimodules if and only if θ is null on F c (1)(V )=V . Examples. The fundamental examples of a quasi-free properad and a quasi-cofree F −1C coproperad are given by the cobar construction Σ and the bar construction F c ΣP . These two constructions are studied in the next section.

4. Bar and cobar construction We generalize the bar and cobar constructions of associative algebras (cf. Sémi- naire H. Cartan [C]) and for operads (cf. E. Getzler, J. Jones [GJ]) to properads. First, we study the properties of the partial composition products and coproducts. We define the reduced bar and cobar constructions and then the bar and cobar constructions with coefficients. These constructions share interesting homological properties. For instance, they induce resolutions for differential properads and props (cf. Theorem 7.8 and Section 7). To show these resolutions, we prove that the augmented bar and cobar constructions are acyclic, at the end of this section.

4.1. Partial composition product and coproduct. We define the notions of suspension and desuspension of a dg-S-bimodule. Then, we give the definitions and first properties of the partial composition products and coproducts.

4.1.1. Suspension and desuspension of a dg-S-bimodule. Let Σ be the graded S-bimodule deﬁned by

Σ(0, 0) = k.Σ where Σ is an element of degree +1, Σ(m, n) = 0 otherewise. Deﬁnition (Suspension ΣV ). The suspension of the dg-S-bimodule V is the dg- S-bimodule ΣV := Σ ⊗ V .

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Taking the suspension of a dg-S-bimodule V corresponds to taking the tensor product of every element of V with Σ. To an element v ∈ Vd−1, one associates s⊗v ∈ (ΣV )d, which is often denoted Σv. Therefore, (ΣV )d is naturally isomorphic to Vd−1. Following the same rules as in the previous section, the diﬀerential on ΣV is given by the formula δ(Σv)=−Σδ(v), for every v in V . The suspension ΣV corresponds to the introduction of an extra element of degree +1. This involves Koszul-Quillen sign rules when permutating objects. Inthesameway,wedeﬁnethedg-S-bimodule Σ−1 by

Σ−1(0, 0) = k.Σ−1 where Σ−1 is an element of degree −1, Σ−1(m, n)=0 otherwise. Deﬁnition (Desuspension Σ−1V ). The desuspension of the dg-S-bimodule V is the dg-S-bimodule Σ−1V := Σ−1 ⊗ V . 4.1.2. Partial composition product. Let (P,µ,η,ε) be an augmented properad. Since 2-level connected graphs are endowed with a bigraduation given by the number of vertices on each level, we can decompose the composition product µ on connected graphs in the following way: µ = r, s ∈N µ(r, s),where ⊕ P ⊕ P → P µ(r, s) :(I ) c (I ) . r s

The product µ(r, s) gives the composition of r non-trivial operations with s non- trivial operations. Definition (Partial composition product). We call a partial composition product ⊕ P ⊕ P the restriction µ(1, 1) of the composition product µ to (I ) c (I ). 1 1 Notice that the partial composition product is the composition product of two non-trivial elements of P where at least one output of the first one is linked to one input of the second one. Remark. When P is an operad, this partial composition product is exactly the partial product denoted ◦i by V. Ginzburg and M.M. Kapranov in [GK]. In the 1 framework of 2 -PROPs introduced by M. Markl and A.A. Voronov in [MV], these authors only consider partial composition products where the upper element has only one output or where the lower element has only one input. The partial composition product defined by W.L. Gan in [G] on dioperads only involves graphs where the two elements of P are linked by a unique edge. In these cases, the related monoidal product is generated by the partial composition product (every composition can be written with a finite number of partial products). One can see that the composition product of a properad has the same property. Lemma 4.1. Let (P,µ,η,ε) be an augmented dg-properad. The partial composition product µ(1, 1) induces a homogenous morphism θ of degree −1: F c P ⊕ P ⊕ P → P θ : (2)(Σ )=(I Σ ) c (I Σ ) Σ . 1 1 Recall that when P is a dg-S-bimodule, the free properad F(ΣP)onΣP is bigraded. The first graduation comes from the number of operations of ΣP used to represent an element of F(ΣP). We denote this graduation F(n)(ΣP). The second

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graduation, the homological degree, is given by the sum of the homological degrees of the operations of ΣP. Proof. Let (1,..., 1, Σq, 1,..., 1)σ(1,..., 1, Σp, 1,..., 1) represent a homogenous | | | | F c P element of homological degree q + p +2 of (2)(Σ ). We deﬁne θ by the following formula: θ ((1,..., 1, Σq, 1,..., 1)σ(1,..., 1, Σp, 1,..., 1)) |q| := (−1) Σµ(1, 1)(1,..., 1,q,1,..., 1)σ(1,..., 1,p,1,..., 1). Since P is a diﬀerential properad, this last element has degree |q| + |p| +1.

c c With Proposition 3.15, we can associate a coderivation dθ : F (ΣP) →F (ΣP) to any homogenous morphism θ.

Proposition 4.2. The coderivation dθ has the following properties :

(1) The equation δdθ + dθδ =0is true if and only if the partial composition product µ(1, 1) is a morphism of dg-S-bimodules. 2 (2) We have dθ =0. Proof. c (1) Following Proposition 3.15, applying dθ to an element of F (ΣP)represented by a graph g corresponds to applying θ to every possible sub-graph of g. Here, we apply θ to every couple of vertices linked by at least one edge and such that there exists no vertex between. For a couple of such vertices indexed by Σq and Σp,wehave δ ◦ d ((1,..., 1, Σq, 1,..., 1)σ(1,..., 1, Σp, 1,..., 1)) θ =(−1)|q|δ Σµ ((1,..., 1,q,1,..., 1)σ(1,..., 1,p,1,..., 1)) (1 , 1) |q|+1 =(−1) Σ δ µ(1, 1)((1,..., 1,q,1,..., 1)σ(1,..., 1,p,1,..., 1)) and ◦ dθ δ((1,..., 1, Σq, 1,..., 1)σ(1,..., 1, Σp, 1,..., 1)) = d − (1,..., 1, Σδ(q), 1,..., 1)σ(1,..., 1, Σp, 1,..., 1) θ +(−1)|q|(1,..., 1, Σq, 1,..., 1)σ(1,..., 1, Σδ(p), 1,..., 1) |q| =(−1) Σµ(1, 1)((1,..., 1,δ(q), 1,..., 1)σ(1,..., 1,p,1,..., 1))

+Σµ(1, 1)((1,..., 1,q,1,..., 1)σ(1,..., 1,δ(p), 1,..., 1)). Therefore, δd + d δ = 0 implies that θ θ δ µ(1, 1)((1,..., 1,q,1,..., 1)σ(1,..., 1,p,1,..., 1))

= µ(1, 1)((1,..., 1,δ(q), 1,..., 1)σ(1,..., 1,p,1,..., 1)) |q| +(−1) µ(1, 1)((1,..., 1,q,1,..., 1)σ(1,..., 1,δ(p), 1,..., 1)),

which means that the partial composition product µ(1, 1) is a morphism of adg-S-bimodule. In the other way, applying δdθ + dθδ corresponds to do a sum of expressions of the previous type. 2 (2) To show that dθ = 0, we have to consider the pairs of couples of vertices of a graph g. There are two possible cases : the pairs of couples of vertices such that the four vertices are diﬀerent (a), the pairs of couples of vertices with one common vertex (b).

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(a) To an element of the form

X ⊗ Σq1 ⊗ Σp1 ⊗ Y ⊗ Σq2 ⊗ Σp2 ⊗ Z, we apply θ twice beginning with one diﬀerent couple each time. It gives | | | | | | | | | | | | | | | | | | | | | | | | | | | | (−1) X + q1 (−1) X + q1 + p1 +1+ Y + q2 +(−1) X + q1 + p1 + Y + q2 (−1) X + q1

×X ⊗ Σµ(1, 1)(q1 ⊗ p1) ⊗ Y ⊗ Σµ(1, 1)(q2 ⊗ p2) ⊗ Z =0. (b) In this case, there are two possible conﬁgurations: • The common vertex is between the two other vertices (following the ﬂow of the graph) (cf. Figure 4 ).

Σp

Σq

Σr

Figure 4.

Therefore, to an element of the form X ⊗ Σr ⊗ Σq ⊗ Σp ⊗ Y, if we apply θ twice beginning by a diﬀerent couple each time, we have |X|+|r| |X|+|r|+|q| (−1) (−1) X ⊗ Σµ(1, 1)(µ(1, 1)(r ⊗ q) ⊗ p) ⊗ Y |X|+|r|+1+|q| |X|+|r| +(−1) (−1) X ⊗ Σµ(1, 1)(r ⊗ µ(1, 1)(q ⊗ p)) ⊗ Y =0,

by associativity of the partial product µ(1, 1) (which comes from the associativity of µ). • Otherwise, the common vertex is above (cf. Figure 5) or under the two others. The same kind of calculus this time gives |X|+|r|+1+|q| |X|+|r| (−1) (−1) X ⊗ Σµ(1, 1)(r ⊗ µ(1, 1)(q ⊗ p)) ⊗ Y (|r|)+1(|q|+1)+|X|+|q|+1+|r| |X|+|q| +(−1) (−1) X ⊗ Σµ(1, 1)(q ⊗ µ(1, 1)(r ⊗ p)) ⊗ Y =0.

Also, the associativity of µ(1, 1) gives |r||q| µ(1, 1)(q ⊗ µ(1, 1)(r ⊗ p)) = (−1) µ(1, 1)(r ⊗ µ(1, 1)(q ⊗ p)). 2 We conclude by remarking that the image under the morphism dθ is a sum of expressions of such type.

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Σp || || || || Σr Σq

Figure 5.

Remark. This proposition lies on the natural Koszul-Quillen sign rules. In the article of [GK], these signs appear via the operad Det. Deﬁnition (Pair of adjacent vertices). Any pair of vertices of a graph linked by at least on edge and without a vertex between is called a pair of adjacent vertices.It F F c corresponds to sub-graphs of the form (2)(V )= (2)(V ) and are the composable pairs under the coderivation dθ.

Remark. The image under the coderivation dθ is given by the composition of all pairs of adjacent vertices. In order to define the homology of graphs, M. Kontsevich has introduced in [Ko] the notion of an edge contraction. This notion corresponds to the composition of a pair of vertices linked by only one edge. In the case of 1 operads (trees), 2 -PROPs and dioperads (graphs of genus 0), adjacent vertices are linked by only one edge. Therefore, this notion applies there. But in the framework of properads, this notion is too restrictive and its natural generalization is given by the coderivation dθ. 4.1.3. Partial coproduct. Here we dualize the results of the previous section. Definition (Partial coproduct). Let (C, ∆,ε,η) be a coaugmented coproperad. The following composition is called a partial coproduct : C −∆→C C ⊕ C¯ ⊕ C¯ c (I ) c (I ). 1 1 It corresponds to the part of the image of ∆ represented by graphs with two vertices. We denote it as ∆(1, 1). Lemma 4.3. Let (C, ∆,ε,η) be a differential coaugmented coproperad. The partial coproduct induces an homogenous morphism of degree −1 −1C→F¯ −1C¯ ⊕ C¯ ⊕ C¯ θ :Σ (2)(Σ )=(I ) c (I ). 1 1 Proof. Let c be an element of C¯. Using the same kind of notations as Sweedler for Hopf algebras, we have ∆(1, 1)(c)= (1,..., 1,c, 1,..., 1)σ(1,..., 1,c , 1,..., 1). (c,c)

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Hence, we deﬁne θ(Σ−1c) by the formula θ(Σ−1c):=− (−1)|c |(1,..., 1, Σ−1 c, 1,..., 1)σ(1,..., 1, Σ−1 c, 1,..., 1). (c,c)

Remark. The sign − appearing in the deﬁnition of θ is not essential here. Its role will be obvious in the proof of the bar-cobar resolution (cf. Theorem 5.8). −1 −1 With Proposition 3.14, we can associate a derivation dθ : F(Σ C¯) →F(Σ C¯) of degree −1 to the homogenous morphism θ which veriﬁes the following properties. Proposition 4.4.

(1) The equation δdθ + dθ δ =0is true if and only if the partial coproduct ∆(1, 1) is a morphism of dg-S-bimodules. 2 (2) We have dθ =0. Proof. The proof of Proposition 3.14 shows that the image of an element of F(Σ−1C¯) represented by a graph g under the derivation dθ is a sum where the application θ is applied on each vertex of g. Therefore, the arguments to show this proposition are the same as for the previous proposition. For instance, the second point comes from the coassociativity of ∆(1, 1), which is induced by the one of ∆, and from the rules of signs.

Remark. The derivation dθ corresponds to taking the partial coproduct of every element indexing a vertex of a graph. Therefore, this derivation is the natural generalization of the notion of vertex expansion introduced by M. Kontsevich in [Ko] in the framework of graph cohomology. In the case of operads, the derivation dθ is the same morphism as the one defined by the “vertex expansion” of trees. 4.2. Definitions of the bar and cobar constructions. With the propositions given above, we can now define the bar and cobar constructions for properads. 4.2.1. Reduced bar and cobar constructions. Definition (Reduced bar construction). To any differential augmented properad (P,µ,η,ε), we can associate the quasi-cofree coproperad B¯(P):=F c(ΣP)with the differential defined by δθ = δ + dθ,whereθ is the morphism induced by the partial product of P (cf. Lemma 4.1). This quasi-cofree coproperad is called the reduced bar construction of P. B¯ P F c P We denote (s)( ):= (s)(Σ ). Therefore, dθ defines the following chain complex:

dθ /¯ dθ /¯ dθ / dθ /¯ dθ /¯ ··· B(s)(P) B(s−1)(P) ··· B(1)(P) B(0)(P).

2 Proposition 4.2 gives that δdθ + dθδ =0andthatdθ = 0. Hence, we have 2 δθ = 0 and we see that the reduced bar construction is the total complex of a bicomplex. Dually, we have the following definition: Definition (Reduced cobar construction). To any differential coaugmented coproperad (C,∆,ε, η) we can associate the quasi-free properad B¯c(C):=F(Σ−1C¯) with the differential defined by δθ = δ + dθ ,whereθ is the morphism induced by

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the partial coproduct of C (cf. Lemma 4.3). This quasi-free properad is called the reduced cobar construction of C.

Once again, the derivation dθ deﬁnes a cochain complex

B¯c C dθ /B¯c C dθ /··· dθ /B¯c C dθ /B¯c C dθ /··· (0)( ) (1)( ) (s)( ) (s+1)( ) . The reduced cobar construction is the total complex of a bicomplex. Remark. When P (or C) is an algebra or an operad (a coalgebra or a cooperad), we find the classical definitions of bar and cobar construction (cf. Séminaire H. Cartan [C] and [GK]). 4.2.2. Bar construction with coefficients. Let (P,µ,η,ε)beanaugmented differential properad. On B¯(P) we can define two homogenous morphisms θr : B¯(P) → B¯(P) c P and θl : B¯(P) →Pc B¯(P)ofdegree−1by B¯ P F c P −∆→Fc P F c P F c P ⊕ P θr : ( )= (Σ ) (Σ ) c (Σ ) (Σ ) c (I ), 1 and by B¯ P F c P −∆→Fc P F c P ⊕ P F c P θl : ( )= (Σ ) (Σ ) c (Σ ) (I ) c (Σ ). 1 Recall that these two morphisms correspond to extract one non-trivial operation from the top or the bottom of graphs that represent elements of F c(ΣP). − Lemma 4.5. The morphism θr induces a homogenous morphism dθr of degree 1: B¯ P P→B¯ P P dθr : ( ) c ( ) c .

Also, the dg-S-bimodule B¯(P) c P endowed with the differential defined by the sum of the canonical differential δ with the coderivation dθ of B¯(P) and the morphism P dθr is an analytic quasi-free -module(ontheright). 2 2 Proof. Since (δ + dθ) = 0, the equation (δ + dθ + dθ ) = 0 is equivalent to r

(δ + dθ)dθr + dθr (δ + dθ)=0 and 2 dθr =0. These two equations are proved in the same arguments as in the proof of Propo- sition 4.2. The functor B¯(P)isanalyticinP, therefore the quasi-free P-module B¯(P) c P is analytic.

The morphism dθr corresponds to taking one by one operations indexing top vertices of the graph of an element of B¯(P) and composing them with operations of P that are on the ﬁrst level of B¯(P) c P. Deﬁnition (Augmented bar construction). Theanalyticquasi-freeP-module B¯(P) c P is called the augmented bar construction (on the right).

One has the same lemma with θl and dθl and deﬁnes the augmented bar con- structionontheleftP c B¯(P). In the case of algebras and operads, the augmented bar construction is acyclic. In the following part of the text, we generalize this result to the augmented bar construction of properads.

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Denote B(P, P, P):=P c B¯(P) c P. The morphisms θr and θl induce two homogenous morphisms P P P →B P P P dθr ,dθl : B( , , ) ( , , ), of degree −1. We define the differential d on B(P, P, P) by the sum of the following morphisms : • the canonical differential δ induced by the one of P, • the coderivation dθ defined on B¯(P), • − the homogenous morphism dθr of degree 1, • − the homogenous morphism dθr of degree 1. Lemma 4.6. The morphism d verifies the equation d2 =0.

Proof. Once again, the proof lies on rules of sign.

Definition (Bar construction with coefficients). Let (P,µ,η,ε)beanaugmented dg-properad, L be a differential right P-module and R be a differential left P- module. The bar construction of P with coefficients in L and R is the dg-S-bimodule B(L, P,R):=LcP B(P, P, P)cP R, where the differential d is induced by the one of B(P, P, P), defined previously, and by the ones of L and R denoted as δR and δL. Proposition 4.7. The bar construction B(L, P,R) with coefficients in L and R is isomorphic, as a dg-S-bimodule, to L c B¯(P) c R, where the differential d is defined by the sum of the following morphisms : • the canonical differential δ induced by the one of P on B¯(P), • the canonical differential δL induced by the one of L, • the canonical differential δR induced by the one of R, • the coderivation dθ defined on B¯(P), • − a homogenous morphism dθR of degree 1, • − a homogenous morphism dθL of degree 1.

Proof. One has to understand the eﬀect of the morphism dθr on the relative com-

position product. The morphism dθr corresponds to the morphism dθR deﬁned by the following composition:

B¯ P θr L c ( ) cr L c B¯(P) c R −−−−→ L c B¯(P) c P c R −−−−−−−−→ L c B¯(P) c R,

where the morphism θr is induced by θr.

From this description of the bar construction with coeﬃcients, we immediately have the following corollary.

Corollary 4.8. • The reduced bar construction is isomorphic to the bar construction with scalar coeﬃcients L = R = I. We have B¯(P)=B(I, P,I). • The chain complex B(L, P, P)=L c B¯(P) c P (resp. B(P, P,R)= P c B¯(P) c R)isananalyticquasi-freeP-module on the right (resp. on the left).

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In the same way as Proposition 3.13 and Theorem 5.9, we have that the bar construction is an exact functor.

Proposition 4.9. The bar construction B(L, P,R) preserves quasi-isomorphisms.

Proof. Let β : P→P be a quasi-isomorphism of augmented dg-properads. Let α : L → L be a quasi-isomorphism of right dg-P,P-modules and let γ : R → R be a quasi-isomorphism of left dg-P,P-modules. Denote by Φ the induced morphism B(L, P,R) →B(L , P ,R). To show that Φ is a quasi-isomorphism, B P we introduce the following filtration Fr := s≤r (s)(L, ,R), where s is the number of vertices of the underlying graph indexed by elements of ΣP.Onecan see that this filtration is stable under the differential d of the bar construction. The canonical differentials δ, δL and δR go from B(s)(L, P,R)toB(s)(L, P,R)andthe B P B P maps dθ, dθR and dθR go from (s)(L, ,R)to (s−1)(L, ,R). Therefore, this ∗ 0 B P filtration induces a spectral sequence Es, t such that Es, t = (s)(L, ,R)s+t and 0 such that d = δ + δL + δR. By the same ideas as in the proof of Proposition 3.13, 0 we have that Es, t(Φ) is a quasi-isomorphism. Since the filtration Fr is bounded below and exhaustive, the result is given by the classical convergence theorem of spectral sequences (cf. J.A. Weibel [W] 5.5.1).

4.2.3. Cobar construction with coefficients. We dualize the previous arguments to define the cobar construction with coefficients. Let (C, ∆,ε,η) be a differential coaugmented coproperad and let (L, l)and (R, r)betwodifferentialC-comodules on the right and on the left. The two C-comodules induce the following morphisms: −→Cr ⊕ C¯ θR : R c R (I ) c R, 1 −→r C ⊕ C¯ θL : L L c L c (I ). 1 Lemma 4.10. The applications θR and θL both induce a homogenous morphism c of degree −1 on L c B¯ (C) c R of the form

θ R / d : B¯c C B¯c C ⊕ C¯ θR L c ( ) c R L c ( ) c (I ) c R 1

LcµB¯c C cR −→ B¯c C ⊕ −1C¯ ( ) / B¯c C L c ( ) c (I Σ ) c R L c ( ) c R , 1

θ c L / θ : d : L B¯ (C) R ⊕ C¯ B¯c C L θL c c L c (I ) c ( ) c R 1

LcµB¯c C cR −→ ⊕ −1C¯ B¯c C ( ) / B¯c C L c (I Σ ) c ( ) c R L c ( ) c R . 1

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c Proposition 4.11. The differential d on the S-bimodule L c B¯ (C) c R is the sum of the following morphisms : • the canonical differential δ induced by the one of C on B¯c(C), • the canonical differential δL induced by the one of L, • the canonical differential δR induced by the one of R, c • the derivation dθ defined on B¯ (C), • the homogenous morphism d of degree −1, θR • the homogenous morphism d of degree −1. θL c Definition (Cobar construction with coefficients). The dg-S-bimodule LcB¯ (C)c R endowed with the differential d is called the cobar construction with coefficients in L and R and is denoted Bc(L, C,R). Corollary 4.12. • The reduced cobar construction B¯c(C) is isomorphic to the cobar construction with coefficients in the scalar C-comodule I. c c • The chain complex B (L, C, C)=L c B¯ (C) c C (resp. B(C, C,R)= c C c B¯ (C) c R)isananalyticquasi-cofreeC-comodule on the right (resp. on the left) 4.3. Acyclicity of the augmented bar and cobar constructions. When P is an algebra A, that’s to say when the dg-S-bimodule P is concentrated in P(1, 1) = A, one has that the augmented bar construction is acyclic. To prove it, one directly introduces a contracting homotopy (cf. Séminaire H. Cartan [C]). In the case of operads, B. Fresse has proved that the augmented bar construction on the left P◦B¯(P) is acyclic using a contracting homotopy. This homotopy lies on the fact the monoidal product ◦ is linear on the left. The acyclicity of the augmented bar construction on the right comes from this result and the use of comparison lemmas for quasi-free modules (cf. [Fr] section 4.6). In the framework of properads, since the composition product c is not linear on the left or on the right, we cannot use the same method. We begin by defining a filtration on the chain complex P c B¯(P),d . This filtration induces a conver- gent spectral sequence. Then, we introduce a contracting homotopy on the chain 0 0 complexes Ep, ∗,d ,forp>0, to show that they are acyclic.

4.3.1. Acyclicity of the augmented bar construction. Let (P,µ,η,ε)bea augmented dg-properad. The dg-S-bimodule P c B¯(P) is the image of the functor c P → (I ⊕ P) c F (ΣP). Since this functor is a sum of polynomial functors, it is a split analytic functor (cf. 2.7). For an element of (P,µ,η,ε), the graduation is given by the number of vertices indexed by elements of P. Once again, we denote this (total) graduation by P B¯ P P B¯ P c ( )= c ( ) (s) . s∈N We consider the ﬁltration deﬁned by P B¯ P P B¯ P Fi = Fi ( ) := c ( ) (s) . s≤i

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Therefore, Fi is composed by elements P B¯(P) represented by graphs indexed at most i elements of P.

Lemma 4.13. The filtration Fi of the dg-S-bimodule P c B¯(P) is stable under the differential d of the augmented bar construction. Proof. Lemma 4.5 gives the form of the differential d of the augmented bar construction. It is the sum of the three following terms : (1) The differential δ coming from the one of P. This one does not change the number of elements of P. Hence, one has δ(Fi) ⊂ Fi. (2) The coderivation dθ of the reduced bar construction B¯(P). This morphism corresponds to the composition of pairs of adjacent vertices. Therefore, it verifies dθ(Fi) ⊂ Fi−1.

(3) The morphism dθl . Applying this morphism corresponds to taking an operation of ΣP of B¯(P) from the bottom and composing it with elements of P on the last level of the graph,

P c B¯(P) →Pc (I ⊕ P) c B¯(P) →Pc B¯(P). P ⊂ The total number of elements of is decreasing, and one has dθl (Fi) Fi.

This lemma shows that the filtration Fi induces a spectral sequence, denoted as ∗ Ep, q. The first term is equal to 0 P B¯ P P B¯ P Ep, q = Fp ( c ( ))p+q /Fp−1 ( c ( ))p+q , 0 where p + q is the homological degree. Hence, the module Ep, q is given by the graphs with exactly p vertices indexed by elements of P.Wehave 0 P B¯ P Ep, q = c ( ) , (p) p+q and the differential d0 is a sum of the morphisms d0 = δ + d . The morphism δ θl is induced by the one of P, and the morphism d is equal to the morphism d θl θl when it does not change the total number of vertices indexed by P. Therefore, the morphism d takes one operation of ΣP in B¯(P) from the bottom and inserts it on θl the line of elements of P (in the last level) without composing it with operations P of . Otherwise, the morphism dθl implies a non-trivial composition with elements P of ,inthiscasedθ is null (cf. figure 6). l ∗ ≥ The spectral sequence Ep, q is concentrated in the half-plane p 0. We compute 0 0 the homology of the chain complexes Ep, ∗,d to show that the spectral sequence 1 collapses at rank Ep, q.

1 Lemma 4.14. At rank Ep, q, we have

I if p = q =0, E1 = p, q 0 otherwise. Proof. When p =0,wehave

I if q =0, E0 = 0,q 0otherwise,

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B w BB ww || BB ww || BB ww || B ww || Σp42 Σp5 2 zz 22 zz 2 zz 22 zz 22 2 Σp3 D 22 zz DD 2 zz DD 2 zz DD 2 zz D Σp13 Σp2 dθ l ~ p1 w p2 B || ww BB || ww BB || ww BB || ww B

Figure 6. Form of the morphism d θl

0 0 and the diﬀerential d is null on the modules E0,q. Hence, the homology of these modules is equal to I if q =0, E1 = 0,q 0otherwise. When p>0, we introduce a contracting homotopy h for each chain of complexes 0 0 Ep,∗,d . Thanks to Proposition 3.2, we can represent an element of P B¯(P)by(p1, ..., pr) σ (b1, ..., bs). When p1 ∈ I, we deﬁne

h ((p1, ..., pr)σ(b1, ..., bs)) = 0, otherwise h is given by the formula

h ((p1, ..., pr)σ(b1, ..., bs))

(|p1|+1)(|p2|+···+|pr |) =(−1) (1, ..., 1,p2, ..., pr)σ (b ,bi+1, ..., bs), ⊗ B¯ P where b = µF(ΣP) (Σp1 (b1, ..., bi)) with b1, ..., bi the elements of ( ) linked via at least one edge to the vertex indexed by p1 in the graphic representative of (p1, ..., pr)σ(b1, ..., bs). This morphism h is homogenous of degree +1 (P→ΣP) P 0 → and does not change the number of elements of . Therefore, we have h : Ep, q 0 Ep, q+1. Intuitively, the application h takes the “ﬁrst” non-trivial operation between the ones of P on the last level, suspends it, and lifts it between the ones of B¯(P). 0 0 Let verify that hd + d h = id. (1) The rules of signs and suspensions show that hδ + δh =0. (2) We have hd + d h = i . We ﬁrst compute d : θl θl d θl d ((p1, ..., pr)σ(b1, ..., bs)) θl | | | | ··· | | | | ··· | | = (−1) p ( pj+1 + + pr + b1 + + bk ) × (p1,..., p,pj+1 ..., pr)σ(b1,..., bk,bk+1,..., bs),

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⊗ where bk+1 = µF(ΣP)(Σp bk+1). Then, we have hd ((p1, ..., pr)σ(b1, ..., bs)) θl | | | | ··· | | | | ··· | | | | | | ··· | | ··· | | = (−1) p ( pj+1 + + pr + b1 + + bk )+( p1 +1)( p2 + + p + + pr ) × (1,..., 1,p2,..., p,pj+1 ..., pr)σ (b ,bi+1,..., bk,bk+1,..., bs). The same calculus for d h gives θl d h ((p , ..., p )σ(b , ..., b )) θl 1 r 1 s =(p , ..., p )σ(b , ..., b ) 1 r 1 s | | | | ··· | | | | ··· | | | | | | ··· | | ··· | | − (−1) p ( pj+1 + + pr + b1 + + bk )+( p1 +1)( p2 + + p + + pr ) × (1,..., 1,p2,..., p,pj+1 ..., pr)σ (b ,bi+1,..., bk,bk+1,..., bs), − and the sign 1 comes from the commutation of p with Σp1. 0 0 Since h is a homotopy, the chain complexes Ep, ∗,d are acyclic for p>0, which means that we have 1 Ep, q =0 forallq if p>0.

We can now conclude the proof of the acyclicity of the augmented bar construction. Theorem 4.15. The homology of the chain complex P c B¯(P),d is given by : ¯ H0 P c B(P) = I, Hn P c B¯(P) =0 otherwise. " P B¯ P Proof. Since the ﬁltration Fi is exhaustive ( c ( )= i Fi) and bounded below (F−1 P c B¯(P) = 0), we can apply the classical theorem of convergence of ∗ spectral sequences (cf. [W] 5.5.1). Hence, we get that the spectral sequence Ep, q converges to the homology of P B¯(P): c 1 ⇒ P B¯ P Ep, q = Hp+q c ( ),d . We conclude using Lemma 4.14. Corollary 4.16. The augmentation morphism

εP ε c Fc(ΣP) / P c B¯(P) I c I = I is a quasi-isomorphism. This last result can be generalized in the following way : We deﬁne the augmentation morphism ε(P, P, P) by the composition P P cεFc(ΣP) c B(P, P, P)=P c B¯(P)c P −−−−−−−−−−−→Pc I c P = P c P P P P = P. Theorem 4.17. Let R be a left diﬀerential P-module. The augmentation morphism ε(P, P,R)

PP ε(P, P, P)P R / B(P, P,R)=P P B(P, P, P) P R P P P P R = R is a quasi-isomorphism.

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Proof. We deﬁne the same kind of ﬁltration based on the number of elements of P indexing the graphs.

Remark. Using the same arguments, we have the same results for the augmented bar construction on the right B¯(P) c P. 4.3.2. Acyclicity of the augmented cobar construction. Dually, we can prove c that the augmented cobar construction B¯ (C) c C is acyclic. But in order to make the spectral sequence converge, we need to work with a weight graded coproperad C.

Theorem 4.18. For every weight graded coaugmented coproperad C, the homology c of the augmented cobar construction B¯ (C) c C,d is given by ¯c H0 B (C) c C = I, c Hn B¯ (C) c C =0 otherwise. c Proof. We define a filtration Fi on B¯ (C) c C,d by B¯c C C Fi = ( ) c (s) . s≥−i This filtration is stable under the differential d : (1) Since the differential δ does not change the number of elements of C¯,we have δ(Fi) ⊂ Fi. c (2) The derivation dθ of the reduced cobar construction B¯ (C) splits into two parts, each element indexing a vertex. Therefore, the global number of vertices indexed by elements of C¯ is raised by 1. The derivation dθ verifies dθ (Fi) ⊂ Fi−1. C (3) The morphism dθr decomposes elements of into two parts and includes on part in B¯c(C). The global number of elements of C¯ indexing vertices is ⊂ increasing. We have dθr (Fi) Fi. ∗ This filtration induces a spectral sequence Ep, q. Its first term is given by 0 c c c B¯ C C − B¯ C C B¯ C C − Ep, q = Fp ( ( ) c )p+q /Fp 1 ( ( ) c )p+q = ( ( ) c )( p) p+q . 0 0 The differential d is the sum of two terms d = δ+d ,whered corresponds to θr θr C¯ dθr when it does not strictly increase the number of elements of . The morphism d takeselementsofC¯ in the first level, desuspends them and includes them in θr B¯c(C). We then show that I if p = q =0, E1 = p, q 0otherwise.

We deﬁne the image under the homotopy h of an element (b1,..., br) σ (c1,..., ¯c ¯c cs)ofB (C) C by ﬁrst decomposing b1 ∈ B (C)(n) : # / ⊗ −1 b1 b1 Σ c , ∈ B¯c C ∈ C¯ −1 where b1 ( )(n−1) and c . If the element Σ c is linked above to elements of I, then we suspend it and include it in the line of elements of C.Usingthesame 0 0 arguments, one can show that hd + d h = id.

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Since the coproperad C is weight graded, one can decompose the chain complex c B¯ (C) c C into a direct sum indexed by the total weight : c c (ρ) B¯ (C) c C = (B¯ (C) C) . ρ∈N

This decomposition is compatible with the ﬁltration Fi and with the homotopy h. Therefore, we have 1 B¯c C C (ρ) Ep, q ( ( ) c ) =0 foreveryp, q when ρ>0and

I if p = q =0, E1 (B¯c(C) C)(0) = p, q c 0otherwise. c (ρ) Since the ﬁltration Fi is bounded on the sub-complex (B¯ (C) c C) , c (ρ) c (ρ) c (ρ) F0 (B¯ (C) c C) =(B¯ (C) c C) and F−ρ−1 (B¯ (C) c C) =0, we can apply the classical theorem of convergence of spectral sequences (cf. [W] ∗ B¯c C C (ρ) 5.5.1). This gives that the spectral sequence Ep, q ( ( ) c ) converges to the c (ρ) homology of (B¯ (C) c C) . We conclude by taking the direct sum on ρ. Remark. Using the same methods, we can show that the augmented bar construction on the left is acyclic.

5. Comparison lemmas In this section we prove comparison lemmas for quasi-free P-modules and quasi- free properads. These lemmas are generalizations to the composition product c of classical lemmas for the tensor product ⊗k. Notice that B. Fresse has given in [Fr] such lemmas for the composition product ◦ of operads. These technical lemmas allow us to show that the bar-cobar construction of a weight graded properad P is aresolutionofP.

5.1. Comparison lemma for quasi-free modules.

5.1.1. Filtration and spectral sequence of an analytic quasi-free module. Let P be a weight graded differential properad P and let L = M c P be an analytic P (α) quasi-free -module on the right, where M =Υ(V ). Denote Md to be the sub- S-bimodule of M composed by elements of homological degree d and global weight α (cf. 3.4). We define the following filtration on L : (¯α) c F (L):= M (¯l, k¯) ⊗S k[S ] ⊗S P (¯j, ¯ı), s d¯ k¯ k,¯ j¯ j¯ e¯ Ξ |d¯|+|α¯|≤s where the direct sum (Ξ) is over the integers m, n and N and the tuples ¯l, k,¯ j,¯ ¯ı.

Lemma 5.1. The filtration Fs is stable under the differential d of L. Proof. The differential d is the sum of three terms.

• The diﬀerential δM , induced by the one of M, decreases the homological ¯ degree |d| by 1. Therefore, we have δM (Fs) ⊂ Fs−1. • The diﬀerential δP , induced by the one of P, decreases the homological degree |e¯| by 1. We have δP (Fs) ⊂ Fs.

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• By deﬁnition of the analytic quasi-free P-modules, one has that the morphism dθ decreases the total graduation |α¯| by at least 1 and the homological ¯ degree |d| by 1. We have dθ(Fs) ⊂ Fs−2.

∗ This filtration gives a spectral sequence Es, t. The first term of this spectral 0 S sequence Es, t is isomorphic to the following dg- -bimodule : s 0 (¯α) c E = M (¯l, k¯) ⊗S k[S ] ⊗S P (¯j, ¯ı). s, t d¯ k¯ k,¯ j¯ j¯ e¯ Ξ r=0 |d¯|=s−r, |e¯|=t+r |α¯|=r Using the same notations as in section 3.2, we have ⎛ ⎞ s 0 0 ⎝ (¯α) P⎠ Es, t = Is−r, t+r M c . r=0 |α¯|=r On can decompose the weight graded differential S-bimodule L with its weight (ρ) L = ρ L . Since this decomposition is stable under the filtration Fs, the spectral ∗ sequence Es,,t is isomorphic to ∗ ∗ (ρ) Es, t(L)= Es, t(L ). ρ∈N By definition of the analytic quasi-free P-modules, the weight of the elements of V is at least 1. Hence, we have ⎛ ⎞ min(s, ρ) $ 0 (ρ) 0 ⎝ (¯α) P (ρ)⎠ Es, t(L )= Is−r, t+r M c L . r=0 |α¯|=r Proposition 5.2. Let L be an analytic quasi-free P-module. The spectral sequence ∗ Es, t converges to the homology of L: ∗ ⇒ Es, t = Hs+t(L, d). 0 0 1 1 The differential d on Es, t is given by δP , and the differential d on Es, t is given by δ . Therefore, we have M ⎛ ⎞ s 2 2 ⎝ (¯α) P⎠ Es, t = Is−r, t+r M c , r=0 |α¯|=r which can be written s 2 (¯α) ∗ ∗ P Es, t = (H (M))d¯ c (H ( ))e¯. r=0 |α¯|=r |d¯|=s−r |e¯|=t−r " Proof. Since the filtration is exhaustive s Fs = L and bounded below F−1 =0, the classical theorem of convergence of spectral sequences (cf. [W] 5.5.1) shows ∗ that Es, t converges to the homology of L. The form of the differentials d0 and d1 comes from the study of the action of 2 d = δM + δP + dθ on the filtration Fs. Also, the formula of Es, t comes from Proposition 3.4.

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5.1.2. Comparison lemma for analytic quasi-free modules. To show the comparison lemma, we need to prove the following lemma ﬁrst. Lemma 5.3. Let Ψ:P→P be a morphism of weight graded connected dg- properads. Let (L, λ) be an analytic quasi-free P-module and (L,λ) be an analytic quasi-free P-module. Denote L¯ = M and L¯ = M as the indecomposable quotients. We consider the action of P on L induced by the morphism Ψ.LetΦ:L → L be a morphism of weight graded diﬀerential P-modules:

(1) The morphism Φ preserves the ﬁltration Fs and leads to a morphism of spectral sequences E∗(Φ) : E∗(L) → E∗(L). (2) Let Φ:¯ M → M be the morphism of dg-S-bimodules induced by Φ.The 0 0 morphism E (Φ) : M c P→M c P is equal to E = Φ¯ c Ψ. Proof.

(1) Let (m1,..., mb) σ (p1,..., pa)beanelementofFs(M c P). It means that |d¯| + |α¯|≤s. Since, Φ is a morphism of P-modules, we have Φ (m ,..., m ) σ (p ,..., p ) =Φ◦ λ (m ,..., m ) σ (p ,..., p ) 1 b 1 a 1 b 1 a = λ (Φ(m1),..., Φ(mb)) σ (Ψ(p1),..., Ψ(pa)) . Denote i i i i i Φ(mi)=(m1,..., mbi ) σ (p1,..., pai ). The application Φ is a morphism of dg-S-bimodules.# Therefore, we have | ¯i| | ¯i| | ¯i|≤ | ¯i|≤|¯| di = d + e which implies d di and #i d d .SinceΦisa P | ¯i|≤| | morphism of weight graded -modules, we have i α α¯ . We conclude that Φ (m1,..., mb) σ (p1,..., pa) ∈ Fs(L ). 0 (2) The morphism Es, t(Φ) is given by the following morphism on the quotients: 0 → Es, t : Fs(Ls+t)/Fs−1(Ls+t) Fs(Ls+t)/Fs−1(Ls+t). 0 | | ≤ Let (m1,..., mb) σ (p1,..., pa)beanelementofEs, t,where α¯ = r s, |d¯| = s − r and |e¯| = t + r.Wehave 0 Es, t(Φ) (m1,..., mb) σ (p1,..., pa) =0 if and only if Φ (m1,..., mb) σ (p1,..., pa) ∈ Fs−1(L ). We have seen that Φ (m1,..., mb) σ (p1,..., pa) = λ (Φ(m1),..., Φ(mb)) σ (Ψ(p1),..., Ψ(pa)) . \ Therefore Φ (m1,..., mb) σ (p1,..., pa) belongs to Fs(Ls+t) Fs−1(Ls+t) if and only if Φ(m )=m = Φ(¯ m ). Consider Φ(m )=(mi ,..., mi ) i i i i 1 bi i i i i σ (p1,..., pai ) where at least one pj does not belong to I. By deﬁnition i of a weight graded properad, the weight of these pj is at least 1. Since Φ preserves the total weight , we have |α¯i| <αi. Finally, the morphism 0 E (Φ) is equal to Φ¯ c Ψ.

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Theorem 5.4 (Comparison lemma for analytic quasi-free modules). With the same notations as in the previous lemma, if two of the following morphisms ⎧ ⎨ Ψ:P→P, ¯ → ⎩ Φ:M M , Φ:L → L, are quasi-isomorphisms, then the third one is also a quasi-isomorphism.

Remark. If the properads are not weight graded and if Φ = Φ¯ c Ψ, when ΦandΨ¯ are quasi-isomorphisms, then Φ is also a quasi-isomorphism. The proof is straight. 0 We have immediately E (Φ) = Φ¯ c Ψ. Since Φ¯ and Ψ are quasi-isomorphisms, we 2 0 1 get that E (Φ) is an isomorphism by Proposition 5.2 (d = δP et d = δM ). Also, with the same proposition, the convergence of the spectral sequence E∗ shows that Φ is a quasi-isomorphism between L and L. Proof. (1) Suppose that Ψ : P→P and Φ:¯ M → M are quasi-isomorphisms. The previous lemma shows that Φ induces a morphism of spectral sequences ∗ ∗ ∗ 0 E (Φ) : E (L) → E (L )andthatE (Φ) = Φ¯ c Ψ. We conclude with the same arguments. (2) Suppose that Ψ : P→P and Φ : L → L are quasi-isomorphisms. We ∗ (ρ) have seen that the spectral sequence Es, t preserves the decomposition L . Moreover, we have min(s, ρ) 2 (ρ) 2 (¯α) c (β¯) E (L )= I M (¯l, k¯) ⊗S k[S ] ⊗S P (¯j, ¯ı) , s, t s−r, t+r d¯ k¯ k,¯ j¯ j¯ e¯ Ξ r=max(0, −t) χ where the second sum (χ)ison|α¯| = r, |β¯| = ρ − r, |d¯| = s − r and |e¯| = t + r.Whens ≥ ρ,wehaveforr = ρ 2 (¯α) c (β¯) I M (¯l, k¯) ⊗S k[S ] ⊗S P (¯j, ¯ı) s−ρ, t+ρ d¯ k¯ k,¯ j¯ j¯ e¯ χ ⎛ ⎞ ⎜ ⎟ 2 ⎝ (¯α)⎠ = Is−ρ, t+ρ Md¯ |α¯|=ρ |d¯|=s−ρ (ρ) H − (M )ift = −ρ, = s ρ 0otherwise. Finally, we have

2 (ρ) − Es, t(L )=0 ift< ρ, 2 (ρ) (ρ) ≥ Es, −ρ(L )=Hs−ρ(M )ifs ρ. We now show by induction on ρ that Φ¯ (ρ) : M (ρ) → M (ρ) is a quasi- (ρ) (ρ) (ρ) isomorphism H∗(Φ¯ ):H∗(M ) → H∗(M ).

For ρ =0,sinceL(0) = M (0),wehaveΦ¯ (0) =Φ(0), which is a quasi- isomorphism. Suppose now that the result is true for r<ρin every (homological) degree ∗ and for r = ρ in degree ∗

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∗ = d. (Since every chain complex is concentrated in non-negative degree, (ρ) we have that Hs(Φ¯ )isalwaysanisomorphismfors<0.) By the preceding lemma, we have ⎛ ⎞ min(s, ρ) 2 (ρ) 2 ⎝ ¯ (¯α) ⎠ Es, t(Φ )= Is−r, t+r Φ c Ψ . r=0 |α¯|=r

2 (ρ) 2 (ρ) → With the induction hypothesis, we get that Es, t(Φ ):Es, t(L ) 2 (ρ) Es, t(L ) is an isomorphism for s

(ρ) −1 (ρ) (ρ) Fs(cone(Φ )) := Fs−1(Σ L ) ⊕ Fs(L ). This ﬁltration induces a spectral sequence which veriﬁes

1 (ρ) 1 (ρ) E∗,t(cone(Φ )) = cone(E∗,t(Φ )).

1 (ρ) The mapping cone of E∗,t(Φ ) veriﬁes the long exact sequence 1 (ρ) 1 (ρ) ···→H cone(E∗ (Φ )) → H E∗ (L ) s+1 ,t s ,t H E1 (Φ(ρ)) s ∗,t 1 (ρ) 1 (ρ) −−−−−−−−−−→ H E∗ (L ) → H cone(E∗ (Φ )) s ,t s ,t → 1 (ρ) →··· Hs−1 E∗,t(L ) which ﬁnally gives the following long exact sequence (ξ):

E2 (Φ(ρ)) ··· / 2 (ρ) / 2 (ρ) s, t / 2 (ρ) Es+1,t(cone(Φ )) Es, t(L ) Es, t(L )

/ 2 (ρ) / 2 (ρ) /··· Es, t(cone(Φ )) Es−1,t(L ) .

We have seen that for all t<−ρ,

2 (ρ) 2 (ρ) Es, t(L )=Es, t(L )=0.

2 (ρ) Therefore, the long exact sequence (ξ)showsthatEs, t(cone(Φ )) = 0 for all s,whent<−ρ. 2 (ρ) Inthesameway,wehaveseenthatEs, t(Φ ) is an isomorphism for s< 2 (ρ) d+ρ. Thanks to the long exact sequence (ξ), we get that Es, t(cone(Φ )) = 0 for all t,whens

2 (ρ) ∞ (ρ) Ed+ρ, −ρ(cone(Φ )) = Ed+ρ, −ρ(cone(Φ )), 2 (ρ) ∞ (ρ) Ed+ρ+1, −ρ(cone(Φ )) = Ed+ρ+1, −ρ(cone(Φ )).

(ρ) Since the ﬁltration Fs of the mapping cone Φ is bounded below and ∗ (ρ) exhaustive, we know that the spectral sequence Es, t(cone(Φ )) converges

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to the homology of this mapping cone. The morphism Φ(ρ) is a quasi- isomorphism. Hence this homology is null and we have the equalities :

2 (ρ) ∞ (ρ) Ed+ρ, −ρ(cone(Φ )) = Ed+ρ, −ρ(cone(Φ )) = 0, 2 (ρ) ∞ (ρ) Ed+ρ+1, −ρ(cone(Φ )) = Ed+ρ+1, −ρ(cone(Φ )) = 0. If we inject these equalities in the long exact sequence (ξ), we show that 2 (ρ) Ed+ρ, −ρ(Φ ) is an isomorphism. 2 (ρ) (ρ) We conclude using the fact that Ed+ρ, −ρ(L )=Hd(M )andthat 2 (ρ) (ρ) 2 (ρ) ¯ (ρ) Ed+ρ, −ρ(L )=Hd(M ). Therefore, Ed+ρ, −ρ(Φ )isequaltoHd Φ : (ρ) (ρ) Hd(M ) → Hd(M ) which is an isomorphism. (3) Suppose that Φ : L → L and Φ:¯ M → M are quasi-isomorphisms. We are going to use the same methods. Since M (0) = I,wegetfromthe relation min(s, ρ) 2 (ρ) 2 (¯α) c (β¯) E (L )= I M (¯l, k¯) ⊗S k[S ] ⊗S P (¯j, ¯ı) , s, t s−r, t+r d¯ k¯ k,¯ j¯ j¯ e¯ Ξ r=max(0, −t) χ where the second direct sum χ is on |α¯| = r, |β¯| = ρ − r, |d¯| = s − r and |e¯| = t + r,thatfors =0 2 (ρ) 2 P(ρ) P(ρ) E0,t(L )=I0,t( t )=Ht( ). Onecanseethat 2 (ρ) Es, t(L )=0, for s<0. Let us show that the morphisms Ψ(ρ) : P(ρ) →P(ρ) are quasi-isomorphisms, by induction on ρ. By deﬁnition of connected properads, we have that P(0) = P(0) = I. (0) Hence, the morphism Ψ = idI is a quasi-isomorphism. Suppose that the result is true for every r<ρ. We show by induction on t that (ρ) (ρ) (ρ) Ht(Ψ ):Ht(P ) → Ht(P ) is an isomorphism. We know that it is true for t<0. Suppose that it is still true for t

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When we inject these two equalities in the long exact sequence (ξ), we get that the morphism

E2 (Φ(ρ)) 2 (ρ) 0,e / 2 (ρ) E0,e(L ) E0,e(L )

2 (ρ) P(ρ) is an isomorphism. We conclude using the equalities E0,e(L )=He( ), 2 (ρ) P(ρ) 2 (ρ) (ρ) E0,e(L )=He( )andE0,e(Φ )=He(Ψ ).

5.1.3. Comparison lemma for analytic quasi-cofree comodules. We can prove the same kind of result for analytic quasi-cofree comodules. Let C be a connected dg-coproperad and let L = M c C be an analytic quasi-cofree C-comodule on the right, where M =Υ(V ). On this chain complex, we define the following filtration: c (β¯) ¯ ¯ ¯ ⊗S S ⊗S C Fs(L):= Md(l, k) k¯ k[ k,¯ j¯] j¯ e¯ (¯j, ¯ı), (Ξ) |e¯|+|β¯|≤s where the direct sum (Ξ) is on the integers m, n and N and on the tuples ¯l, k¯,¯j, ¯ı, d¯,¯e and β¯ such that |e¯| + |β¯|≤s. Applying the same arguments to this filtration and the associated spectral sequence, we get the following theorem. Theorem 5.5 (Comparison lemma for analytic quasi-cofree comodules). Let Ψ: C→C be a morphism of the weight graded connected dg-coproperad and let (L, λ) and (L,λ) be two analytic quasi-cofree comodules on C and C.DenoteL¯ = M and L¯ = M as the indecomposable quotients. Let Φ:L → L be a morphism of analytic C-comodules, where the structure of C-comodule on L is given by the functor Ψ.DenoteΦ:¯ M → M ⎧as the morphism of dg-S-bimodules induced by Φ. ⎨ Ψ:C→C, ¯ → If two of the three morphisms ⎩ Φ:M M, are quasi-isomorphisms, then Φ:L → L, the third one is a quasi-isomorphism. 5.1.4. Bar-cobar resolution. In this section, we use the two last theorems to prove that the bar-cobar construction on a weight graded properad P is a resolution of P. Notice that this result is a generalization to properads of one given by V. Ginzburg and M.M. Kapranov for operads in [GK]. We define the morphism ζ : B¯c(B¯(P)) →Pby the following composition: B¯c B¯ P F −1F¯c P //F −1F¯c P F P cP /P ( ( )) = Σ (Σ ) Σ (1)(Σ ) = ( ) ,

where the morphism cP corresponds to the compositions µ of all the elements of P that are indexing a graph of F(P). Proposition 5.6. The morphism ζ is a morphism of weight graded diﬀerential properads. Proof. Since the deﬁnition of ζ is based on the composition µ of P, one can see ◦ ◦ that ζ µF(Σ−1F¯c(ΣP)) = µP (ζ c ζ). Hence, the morphism ζ is a morphism of properads.

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The morphism ζ is either null or given by compositions of operations which preserves the weight. Therefore, the morphism ζ is a morphism of weight graded properads. It remains to show that ζ commutes with the differentials. On B¯c(B¯(P)) = F Σ−1F¯c(ΣP) the differential d is the sum of the derivation c dθ of the cobar construction induced by the partial coproduct of F (ΣP)with the canonical differential δB¯(P). Also, the canonical differential δB¯(P) is the sum of B¯ P the coderivation dθ of the bar construction ( ) induced by the partial product −1 c of P with the canonical differential δP .Letξ be an element of F Σ F¯ (ΣP) represented by an indexed graph g. Three different cases are possible. • F c P P When all the vertices of g are indexed by elements of (1)(Σ )=Σ ,the image of ξ under the differential d is equal to the image under the canonical differential δP . Since the composition µ of P is a morphism of differential properads, we have ζ ◦ d(ξ)=ζ ◦ δP (ξ)=δP ◦ ζ(ξ). • F c P When at least one vertex of g is indexed by an element of (s)(Σ ), with s ≥ 3, the image of ξ under ζ is null. Moreover, the image of ξ under the differential d is a sum of graphs where at least one vertex is indexed by an F c P ≥ ◦ ◦ element of (s)(Σ ), with s 2. Therefore, we have d ζ(ξ)+ζ d(ξ)=0. • F c P When the vertices of g are indexed by elements of (s)(Σ )wheres =1, 2, F c P with at least one of (2)(Σ ), the image of ξ under ζ is null. It remains to c show that ζ ◦ d(ξ) = 0. Since δP preserves the natural graduation of F , one has ζ ◦ δP (ξ) = 0. Study the image under dθ + dθ of an element of F c P ⊗ −1 ⊗ ⊗ ⊗ (2)(Σ ). We denote ξ = X Σ (Σp1 Σp2) Y ,where(Σp1 Σp2) − F c P F F¯c P belongs to (2)(Σ )andX, Y to (Σ 1 (Σ )). Applying dθ ,weget one term of the form

|X|+1 |p1|+1 −1 −1 (−1) (−1) X ⊗ Σ Σp1 ⊗ Σ Σp2 ⊗ Y, −1 −1 −1F c P where Σ Σp1 and Σ Σp2 are elements of Σ (1)(Σ ). Also, applying dθ,wegetonetermoftheform

|X|+1 |p1| −1 (−1) (−1) X ⊗ Σ Σµ(p1 ⊗ p2) ⊗ Y, − − 1 ⊗ 1F c P ◦ where Σ Σµ(p1 p2) belongs to Σ (1)(Σ ). Therefore, ζ (dθ +dθ )(ξ) is a sum of terms of the form |X|+|p1| |X|+|p1|+1 (−1) +(−1) X ⊗ µ(p1 ⊗ p2) ⊗ Y =0.

Remark. With this proof, one can understand the use of the sign −1 in the deﬁnition of the derivation dθ of the cobar construction. Denote C := B¯(P) as the connected dg-coproperad deﬁned by the bar construc- c tion on P. We consider the augmented cobar construction on the right B¯ (C)c C = c c B¯ (B¯(P)) c B¯(P). Remark that the coproperad B¯(P)=F (ΣP)isweightgraded c (by the global weight) and that the augmented cobar construction B¯ (C) c C is an analytic quasi-cofree C-comodule.

c Lemma 5.7. The morphism ζ c B¯(P):B¯ (B¯(P)) c B¯(P) →Pc B¯(P) between c the augmented cobar construction B¯ (C) c C and the augmented bar construction

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P c B¯(P) is a morphism of dg-S-bimodules. Moreover, it is a morphism of analytic quasi-free B¯(P)-comodules. S Proof. Since ζ and idB¯(P) are morphisms of dg- -bimodules, the morphism ζ c idB¯(P) preserves the canonical diﬀerentials δB¯c(C) + δC and δP + δB¯(P). One can see

that the morphism dθr that appears in the definition of the differential of the cobar B¯c C C B¯ P construction ( ) c (cf. 4.2) corresponds via ζ c ( ) to the morphism dθl of P B¯ P the differential of the bar construction c ( ). (The morphism dθr corresponds to extract an operation of P from C = B¯(P) from the bottom and compose it, from B¯c B¯ P P the top, in ( ( )). The morphism dθl corresponds to extract an operation of in B¯(P) from the bottom to compose it at the top of P.) P Since ζ and idB¯(P) preserve the weight graduation coming from the one of ,we have that ζ c B¯(P) is a morphism of analytic quasi-free B¯(P)-comodules. Theorem 5.8 (Bar-cobar resolution). For every connected dg-properad P,themor- phism ζ : B¯c(B¯(P)) →P is a quasi-isomorphism of weight graded dg-properads. Proof. We know from Theorem 4.18 that the augmented cobar construction c B¯ (B¯(P)) c B¯(P) is acyclic and from Theorem 4.15 that the augmented bar construction P c B¯(P) is acyclic too. Therefore, the morphism ζ c B¯(P)isaquasi- isomorphism. We apply the comparison lemma for analytic quasi-cofree comodules B¯ P (Theorem 5.5) to the quasi-isomorphisms Ψ = idB¯(P) and Φ = ζ c ( ). It gives that ζ = Φ¯ is a quasi-isomorphism.

Remark. In the rest of the text, we will mainly apply this theorem to quadratic properads (which are naturally graded). The comparison lemma for quasi-free properads, proved in the next section, will allow us to simplify the bar-cobar resolution to get a quasi-free resolution, namely the Koszul resolutions (cf. Section 7). 5.2. Comparison lemma for quasi-free properad. Here we show a comparison lemma of the same type but for quasi-free properads. Theorem 5.9 (Comparison lemma for quasi-free properads). Let M and M be two weight graded dg-S-bimodules such that M (0) = M (0) =0.LetP and P be two quasi-free properads of the form P = F(M) and P = F(M ), endowed with derivations dθ and dθ coming from decomposable morphisms of weight graded dg- S → F → F -bimodules θ : M s≥2 (s)(M) and θ : M s≥2 (s)(M ). Given a morphism of weight graded dg-properads Φ:P→P, we consider the induced ¯ morphism Φ:M = F(1)(M) → M = F(1)(M ). The morphism Φ is a quasi-isomorphism if and only if Φ¯ is a quasi-isomorphism. Proof. Since the morphisms θ and θ preserve the weight, we have that the derivations dθ and dθ preserve the weight. Therefore, the diﬀerentials δθ = δM + dθ and δθ = δM + dθ preserve the weight, and we can decompose the chain complexes F F (ρ) F F (ρ) (M)= ρ∈N (M) and (M )= ρ∈N (M ) with the weight. We introduce the following ﬁltration: Fs(F(M)) := F(r)(M). r≥−s

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This ﬁltration is compatible with the preceding decomposition. To make the spectral sequences converge, we consider the sub-complexes F(M)(ρ) and the ﬁltration (ρ) (ρ) Fs(F(M) )= F(r)(M) . r≥−s Since the weight graded dg-S-bimodules M and M are concentrated in weight > 0, (ρ) (ρ) we have F(r)(M) = F(r)(M ) =0whenr>ρand (ρ) (ρ) Fs(F(M) )= F(r)(M) . −s≤r≤ρ

One can see that the filtration Fs is stable under the differential δθ = δM + dθ. (We have δM (Fs) ⊂ Fs and dθ(Fs) ⊂ Fs−1.) Therefore, this filtration induces a ∗ spectral sequence Es, t. The first term of this spectral sequence is given by the formula 0 F (ρ) F (ρ) F (ρ) Es, t = Fs( (M)s+t)/Fs−1( (M)s+t)= −s(M)s+t, 0 and the differential d is equal to the canonical differential δM . From this descrip- 0 F ¯ tion and the properties of Φ, we get that Es, t(Φ) = (−s)(Φ)s+t. (ρ) The filtration Fs of F(M) is bounded below and exhaustive (F−ρ−1 =0and (ρ) F0 = F(M) ). Hence, the classical convergence theorem of spectral sequences ∗ (cf. [W] 5.5.1) shows that the spectral sequence Es, t converges to the homology of F(M)(ρ). We can now prove the equivalence : (⇐)IfΦ:¯ M → M is a quasi-isomorphism, since the functor F is exact (cf. Proposition 3.13), we have that F ¯ 0 (ρ) 0 F (ρ) → 0 F (ρ) (−s)(Φ)s+t = Es, t(Φ ):Es, t( (M) ) Es, t( (M ) ) is a quasi-isomorphism. Hence, E1(Φ(ρ)) is an isomorphism. The convergence of the spectral sequence shows that Φ(ρ) : F(M)(ρ) →F(M )(ρ) is a quasi-isomorphism.

(⇒) In the other way, if Φ is a quasi-isomorphism, then each Φ(ρ) is a quasi- isomorphism. We will show by induction on ρ that Φ¯ (ρ) : M (ρ) → M (ρ) is a quasi-isomorphism. For ρ =0,wehaveM (0) = M (0) = 0 and the morphism Φ¯ (0) is a quasi- isomorphism. Suppose that the result is true up to ρ and show it for ρ + 1. In this case, since 0 F (ρ+1) F (ρ+1) 1 (ρ+1) Es, t( (M) )= (−s)(M)s+t , one can see that Es, t(Φ ) is an isomorphism for every t,solongass<−1. Once again, we consider the mapping cone of the application Φ(ρ+1). We define the same filtration on the mapping cone cone(Φ(ρ+1))asonF(M)(ρ+1) : (ρ+1) −1 (ρ+1) (ρ+1) Fs(cone(Φ )) := Fs(Σ F(M) ) ⊕ Fs(F(M ) ). This filtration induces a spectral sequence such that 0 (ρ+1) 0 ρ+1 Es, ∗(cone(Φ )) = cone(Es, ∗(Φ )). As in the proof of Theorem 5.4, we have the following long exact sequence: ··· / 1 F (ρ+1) / 1 F (ρ+1) / 1 (ρ+1) Es, t+1( (M) ) Es, t+1( (M ) ) Es, t(cone(Φ ))

/ 1 F (ρ+1) / 1 F (ρ+1) / 1 (ρ+1) /··· Es, t( (M) ) Es, t( (M ) ) Es, t−1(cone(Φ )) .

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1 (ρ+1) − Since Es, t(Φ ) is an isomorphism, for s< 1, the long exact sequence shows 1 (ρ+1) − that Es, t(cone(Φ ) = 0 for every t so long as s< 1. The form of the ﬁltration 1 (ρ+1) ≥ gives that Es, t(cone(Φ )) = 0 for every t when s 0. The spectral sequence ∗ (ρ+1) 1 − Es, t(cone(Φ )) collapses at rank E (only the row s = 1 is not null). It implies that ∞ (ρ+1) 1 (ρ+1) E−1,t(cone(Φ )) = E−1,t(cone(Φ )). Applying the classical convergence theorem of spectral sequences, we get that the ∗ (ρ+1) spectra sequence Es, t(cone(Φ )) converges to the homology of the mapping cone of Φ(ρ+1) which is null since Φ(ρ+1) is a quasi-isomorphism. Therefore we have 1 (ρ+1) E−1,t(cone(Φ )) = 0, for every t, which injected in the long exact sequence gives that 0 (ρ+1) ¯ (ρ+1) (ρ+1) → (ρ+1) E−1,t(Φ )=Φ : M M is a quasi-isomorphism.

Remark. This theorem generalizes to properads a result of B. Fresse [Fr] for “connected” operads (P(0) = 0 and P(1) = k). This hypothesis ensures the convergence of the spectral sequence introduced by the author. To be able to generalize this result, we have introduced the graduation by the weight. Therefore, Theorem 5.9 can be applied to weight graded operads non-necessarily “connected” in the sense of [Fr], for instance operads with unitary operations or weight graded algebras. Corollary 5.10. Let P = F(M) be a quasi-free properad verifying the hypothesis of the previous theorem. The natural projection B¯(F(M)) → ΣM is a quasi- isomorphism. Proof. Denote M := Σ−1B¯(F(M)). The bar-cobar construction of the weight graded connected dg-properad P is a quasi-free properad of the form BcB(F(M)) = F(M ), which is quasi-isomorphic to P = F(M) by Theorem 5.8. The comparison lemma for a quasi-free properad (Theorem 5.9) concludes the proof.

6. Simplicial bar construction In this section, we generalize the simplicial bar construction of associative algebras and operads to properads. We show that the simplicial bar construction of a properad is quasi-isomorphic to its diﬀerential bar construction.

6.1. Deﬁnitions. In every monoidal category (A, 2,I), one can associate to every monoid M a simplicial chain complex on the objects M 2n,forn ∈ N.Inthe category of k-modules, one gets the classical bar construction of associative algebras. In the case of monads, one ﬁnds the triple bar construction of J. Beck [B]. We apply this construction to the monoidal category of S-bimodules.

6.1.1. Simplicial bar construction with coefficients. Definition (Simplicial bar construction with coefficients). Let P be a differential properad and let (L, l)and(R, r)betwodifferentialP-modules on the right and on the left. We denote C P n P ··· P n(L, ,R):=Σ L c c c cR. n

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We deﬁne the face maps di : Cn(L, P,R) →Cn−1(L, P,R)by • the right action l if i =0, • the composition µ of the ith with the (i +1)th line, composed by elements of P, • the left action r if i = n.

The degeneracy maps sj : Cn(L, P,R) →Cn+1(L, P,R)aregivenbythein- j n−j sertion of the unit η of the properad : L c P c c η c P c c R.

The differential dC on C(L, P,R) is defined by the sum of the simplicial differential with the canonical differentials n i+1 dC := δL + δP + δR + (−1) di. i=0 2 n One can check that dC = 0 thanks to the suspension Σ in the definition of the Cn(L, P,R). This chain complex is called the simplicial bar construction with coefficients in L and R. Remark. When the S-modules P, L and R are concentrated in degree 0, this construction is strictly simplicial. Like the differential bar construction (cf. Proposition 4.9), the simplicial bar construction is an exact functor. Proposition 6.1. The simplicial bar construction C(L, P,R) preserves quasi- isomorphisms.

Proof. We use the same proof as for Proposition 4.9 with the ﬁltration Fr := C P s≤r (s)(L, ,R), where s is the number of vertices of the underlying level-graph indexed by elements of P (including the trivial vertices indexed by I).

We define the augmentation morphism ε : C(L, P,R) → LcP R by the canonical projection C0(L, P,R)=L c R LcP R. As in the case of the differential bar construction, one has a notion of reduced simplicial bar construction. Definition (Reduced simplicial bar construction). The simplicial bar construction with trivial coefficients C(I, P,I) is called the reduced simplicial bar construction. We denote it by C¯(P). Remark. We have previously seen that the elements of the bar construction B(L, P,R) can be represented by graphs and that the coderivation dθ corresponds to the notion of “adjacent pair composition”. The elements of the simplicial bar construction can be represented by level graphs, and the faces di correspond to compositions of two levels of operations. 6.1.2. Augmented simplicial bar construction. Definition (Augmented simplicial bar construction). The chain complexes

C(I, P, P)=C¯(P) c P

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and C(P, P,I)=P c C¯(P) are called augmented simplicial bar construction on the right and on the left. Proposition 6.2. For every properad P and every right P-module L, the augmentation morphism ε : C(L, P, P) → L is a quasi-isomorphism of right simplicial P-modules. We have the same result on the left for every left P-module R.

Proof. The proof is the same as in the case of associative algebras (cf. Séminaire Cartan [C]). We consider the following filtration: C P P Fr := (L, , ), ≤ s r s where s denotes the total homological degree of elements of P.Thisfiltration ∗ is stable under the differential dC. It induces a spectral sequence Es, t such that 0 C P P 0 Es, t = (L, , )s+t and d = δL + d. We introduce an extra degeneracy map s sn+1 :

ΣnL Pcn+1 η n cn n cn+1 c c n+1 cn+1 Σ Lc P c PΣ Lc P c I −−−−−−−−−−−−→ Σ Lc P c P, which induces a contracting homotopy on E0. Once again, we conclude with the classical convergence theorem for spectral sequences. Corollary 6.3. The augmented simplicial bar construction on the left and on the right are acyclic. 6.1.3. Normalized bar construction. To any simplicial chain complex, one can associate its normalized chain complex, which is the quotient by the images of the degeneracy maps. Deﬁnition (Normalized bar construction). The normalized bar construction is given by the quotient of the simplicial bar construction by the images of the degen- eracies maps. We denote Nn(L, P,R) as the following dg-S-module: # n−1 ci c(n−i−1) LcP cηcP cR n c(n−1) i=0 cn Σ coker L c P c R −−−−−−−−−−−−−−−−−−−−−−−−−→ L c P c R .

The chain complex N (I, P,I), denoted N¯ (P), is called the reduced normalized bar construction. Remark. The normalized bar construction N (L, P,R) is isomorphic to the sub- complex of the simplicial bar construction C(L, P,R) spanned by the level-graphs indexed by at least one element of P on each level. Proposition 6.4. The normalized bar construction N (L, P,R) preserves quasi- isomorphisms. Proof. The proof is the same as Proposition 6.1.

We compute the homology of the reduced normalized bar construction of a quasi- free properad as we did for the bar construction in Corollary 5.10.

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Proposition 6.5. Let P = F(M) be a quasi-free properad on a weight graded dg- S (0) -bimodule M such that M =0and where the derivation dθ is decomposable, → F N¯ F → θ : M s≥2 (s)(M). The natural projection ( (M)) ΣM is a quasi- isomorphism. ¯ (ρ) Proof. On the sub-complex N (F(M)) , composed by elements of total weight N¯ F (ρ) (ρ), we consider the filtration Fr := s≥−r (s)( (M)) ,wheres denotes the number of vertices indexed by elements of M. The differential dN = d + δM + dθ ∗ 0 preserves this filtration, which induces a spectral sequence Es, t.WehaveEs, t = N¯ F (ρ) 0 0 N¯ F (ρ) N¯ 0 F (ρ) (s)( (M))s+t and d = d+δM . Therefore, E ( ( (M)) )= (E ( (M))) , and the lemma is true if it is true for a weight graded free dg-properad. P F If = (M) is a weight graded free dg-properad, consider the filtration Fr := N¯ F s s≤r ( (M)) ,wheres denotes the sum of the homological degree of the elements of M. One can see that the filtration is stable under the differential 0 N¯ F 0 dN = d + δM .WehaveEs, t = t( (M))s+t and d = d. For every s,wede- 0 fine a contracting homotopy h on Es, ∗, which concludes the proof. Let ξ be an element of N¯t(F(M)) represented by a graph g with t levels. Consider the operations of F¯(M) indexing the last level and for each of them, consider the operations of M that have no operation below (such that there are no outgoing edges attached to another operation). Call such operations extremal. Every outgoing edge of an extremal operation is an outgoing edge of g.Denotem as the extremal operation of ξ that has among its outgoing edges the one indexed by the smallest integer. We define the image of ξ under h by the level graph with t + 1 levels indexed by m on the last level and by the other operations of ξ on the first levels (except the one without m). 6.2. Levelization morphism. The levelization morphism is an injective morphism between the bar construction and the normalized bar construction which induces an isomorphism in homology. B¯ P F c P 6.2.1. Definition. Let ξ be an element of (n)( )= (n)(Σ ) represented by a graph gξ with n vertices (cf. Figure 7).

______ν1 C ν2 CC {{ CC{ { CC }{{ C! Ñ ______µ1 µ2

Figure 7. An example of gξ.

The element ξ comes from a graph gr with r levels under the equivalence relation ≡ (cf. [V1]). Recall that the relation ≡ allows us to change the level of an operation, up to sign. With this equivalence relation, one can represent ξ with at least one

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n-level-graph with n non-trivial vertices (that’s to say, with one non-trivial vertex by level (cf. Figure 8)). Denote Gn(ξ) as the set of graphs with n levels such that there is only one non-trivial vertex on each level, which give gξ under the equivalence relation ≡.

______ν2 _ _ _ _ _ _ _ _ _ _ ν1 C CC CC CC C! Ö _ _ _ _ _ _ _ _ _ _ µ2 Ù ______µ1

Figure 8. An example of g ∈G4(ξ).

Let g be a graph of Gn(ξ). The element ξ is determined by g and by the sequence of operations Σp1 ⊗···⊗Σpn which index the vertices of g. To associate an element of N¯ (P)fromξ, one needs to permutate the suspensions. This makes signs appear. We denote # n (n−i)|pi| n g(ξ):=(−1) i=1 Σ g(p1 ⊗···⊗pn),

where g(p1 ⊗···⊗pn) corresponds to the graph g whose vertices are indexed by the operations p1, ..., pn. Definition (Levelization morphism). We define the levelization morphism e : B¯(P) → N¯ (P) # by the formula e(ξ):= g(ξ), for ξ in B¯ (P). g∈Gn(ξ) (n) 6.2.2. Homological properties. We prove that the levelization morphism is an injective quasi-isomorphism of dg-S-bimodules. Lemma 6.6. Let P be a differential augmented properad. The levelization morphism is an injective morphism of dg-S-bimodules e : B¯(P) → N¯ (P).

Proof. Denote dB as the diﬀerential of B¯(P). It is the sum of 2 terms : dB = δP +dθ. Call dN the diﬀerential on N¯ (P). It is the sum of 2 terms. The image of an n element τ =Σ τ of N¯ (P) under dN is n−1 n n i+1 n−1 dN (τ )=(−1) Σ δP (τ)+ (−1) Σ di(τ). i=1

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Show that dN ◦ e(ξ)=e ◦ dB(ξ).

• The commutativity of the canonical diﬀerentials δP ◦e(ξ)=e◦δP (ξ)comes from the good choice of signs and permutation of suspensions. • The coderivation dθ composes pairs of adjacent operations of ΣP of the graph representing ξ.Also,di ◦ e corresponds to the composition of 2 levels of operations of P. Two cases must be distinguished. Denote ξ = X ⊗ Σp ⊗ Σq ⊗ Y . (1) If the operations# Σp and Σq are linked by at least an edge, then the n−1 − i+1 ◦ part of i=1 ( 1) di e(ξ) involving the composition of p with q is of the form ε(−1)j+2Σn−1X ⊗ µ(p ⊗ q) ⊗ Y ,

where X = p1 ⊗···⊗pj, Y = q1 ⊗···⊗qn−j−2 ⊗ ρ1 ⊗···⊗ρr and

# # − − j (n−i)|p |+(n−j−1)|p|+(n−j−2)|q|+ n j 2(n−j−k−2)|q | ε =(−1) i=1 i k=1 k .

Also, the part of dθ(ξ) involving the composition of Σp with Σq is of the form (−1)|X|+|p|X ⊗ Σµ(p ⊗ q) ⊗ Y. The image of such an element under e is

(−1)|X|+|p|ε(−1)|X |+|p|Σn−1X ⊗ µ(p ⊗ q) ⊗ Y = ε(−1)jΣn−1X ⊗ µ(p ⊗ q) ⊗ Y .

(2) If the operations Σp and Σq are not linked# by an edge, then dθ does n − i+1 ◦ not compose Σp with Σq. Also, the part of i=0( 1) di e(ξ)that comes from the composition of the levels where p and q lie is the sum of the 2 following terms : j n (|p|+1)(|q|+1)+|p|+|q| (−1) Σ dj+1 εX ⊗ p ⊗ q ⊗ Y + ε(−1) X ⊗ q ⊗ p ⊗ Y = (−1)jΣn εX ⊗ p ⊗ q ⊗ Y +ε(−1)(|p|+1)(|q|+1)+|p|+|q|+|p||q|X ⊗ p ⊗ q ⊗ Y =0. # n−1 − i+1 ◦ − We conclude by remarking that the image of ξ under i=1 ( 1) di e e ◦ dθ is a sum of terms of the form (1) or (2). B¯ P F c P Now show the injectivity of e.Letξ be an element of (n)( )= (n)(Σ ). By definition of the cofree (connected) coproperad, one knows that ξ = ξ1 + ···+ ξr F c P where each ξi is a finite sum of elements of (n)(Σ ) coming from indexing the N¯ P → B¯ P vertices of the same graph gξi . We introduce a morphism π : n( ) (n)( ). To any element τ of N¯n(P) represented by a graph with n levels, we associate its image π(τ) under the equivalent relation ≡. Therefore, we have π ◦ e(ξ)=n1ξ1 + ···nrξr where the ni are integers > 0. The equation e(ξ)=0givesn1ξ1 +···+nrξr =0.By identification of the underlying graphs, we have ξi = 0, for all i, hence ξ =0.

The following theorem answers a question raised by M. Markl and A.A. Voronov in [MV] and generalizes a result of B. Fresse for operads in [Fr].

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Theorem 6.7.1 Let P be a weight graded augmented dg-properad. The levelization morphism e : B¯(P) → N¯ (P) is a quasi-isomorphism. Proof. Since the reduced bar construction (cf. Proposition 4.9) and the reduced simplicial bar construction (cf. Proposition 6.1) preserve quasi-isomorphisms, it is enough to prove this theorem for a resolution of P. The properad P is weight graded and augmented. Therefore, Theorem 5.8 shows that the bar-cobar construction B¯c(B¯(P)) of P is a quasi-free resolution of P.DenoteM =Σ−1B¯(P) and P = B¯c(B¯(P)) = F(M). Corollary 5.10 shows that B¯(P)=B¯(F(M)) is quasi-isomorphic to ΣM, and Proposition 6.5 shows that N¯ (P)=N¯ (F(M)) is quasi-isomorphic to ΣM. We have the following commutative diagram of quasi- isomorphism: B¯ P e /N¯ P (O) (O)

q.i. q.i.

B¯(F(M)) e /N¯ (F(M))

q.i. q.i. ΣM ΣM.

Remark. Using the same theorem for operads (cf. [Fr]), the author has proved in [V3] a correspondence between operads and posets (partially ordered sets). To a set operad P, one can associate a family of posets ΠP of partition type. The main theorem of [V3] asserts that the operad is Koszul over Z if and only if each poset of ΠP is Cohen-Macaulay. It seems that the same kind of work can be done in the case of a properad with the previous theorem.

7. Koszul duality We deduce the Koszul duality theory of properads from the comparison lemmas of the previous section. 7.1. Koszul dual. To a weight graded properad P, we deﬁne its Koszul dual as a sub-coproperad of its reduced bar construction. Dually, to a weight graded coproperad C, we deﬁne its Koszul dual as a quotient properad of its reduced cobar construction.

7.1.1. Koszul dual of a properad. Let P be a weight graded augmented dg- properad. Recall that the reduced bar construction B¯(P)=F c(ΣP)ofP is bigraded by the number (s) of non-trivial indexed vertices and by the total weight (ρ) ¯ ¯ (ρ) B(s)(P)= B(s)(P) . ρ∈N

1The proof of this theorem was given to the author by Benoit Fresse.

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Since the coderivation dθ, induced by the partial product, composes pairs of adjacent vertices, we have ¯ (ρ) ¯ (ρ) dθ B(s)(P) ⊂ B(s−1)(P) . When P is connected (P(0) = I), the bar construction of P has the following form. Lemma 7.1. Let P be a weight graded connected dg-properad. We have the equalities B¯ P (ρ) F c P(1) (ρ)( ) = (ρ)(Σ ), ¯ (ρ) B(s)(P) =0 if s>ρ. Remark. The same result holds for the reduced cobar construction of a weight graded connected dg-coproperad. Deﬁnition (Koszul dual of a properad). Let P be a weight graded connected dg-properad. We deﬁne the Koszul dual of P by the weight graded dg-S-bimodule P¡ B¯ P (ρ) (ρ) := H(ρ) ∗( ) ,dθ . (ρ) The preceding lemma gives the form of the chain complex B¯∗(P) ,dθ :

··· dθ / dθ /¯ (ρ) dθ /¯ (ρ) dθ /··· 0 B(ρ)(P) B(ρ−1)(P) . Hence, we have the formula P¡ B¯ P (ρ) → B¯ P (ρ) (ρ) =ker dθ : (ρ)( ) (ρ−1)( ) . P¡ S This formula shows that (ρ) is a weight graded dg- -bimodule, where the differential is induced by the one of P : δP . If the properad is concentrated in homological degree 0, we have

B¯ (P)(ρ) if d = s, B¯ (P)(ρ) = (s) (s) d 0otherwise. The dual coproperad is not necessarily concentrated in degree 0. It veriﬁes

P¡ if d = ρ, P¡ = (ρ) (ρ) d 0otherwise. 7.1.2. Koszul dual of a coproperad. Inthesameway,wedefinetheKoszuldual of a coproperad. Definition (Koszul dual of a coproperad). Let C be a weight graded connected dg- coproperad. We define the Koszul dual of C by the weight graded dg-S-bimodule ¡ C B¯c C (ρ) (ρ) := H(ρ) ∗( ) ,dθ .

The cobar construction endowed with the derivation dθ is a chain complex of the following form:

d d d ··· θ /¯ (ρ) θ /¯ (ρ) θ / B(ρ−1)(C) B(ρ)(C) 0 . Therefore, the Koszul dual of a coproperad is isomorphic to ¡ C B¯c C (ρ) → B¯c C (ρ) (ρ) =coker dθ : (ρ−1)( ) (ρ)( ) .

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C¡ S The module (ρ) is a weight graded dg- -bimodule, where the diﬀerential is induced by the one of C : δC. Proposition 7.2. Let P be weight graded connected dg-properad. The Koszul dual of P,denotedP¡, is a weight graded sub-dg-coproperad of F c(ΣP(1)). Let C be a weight graded connected dg-coproperad. The Koszul dual of C,denoted C¡, is a weight graded dg-properad quotient of F(Σ−1C(1)). P¡ B¯ P (ρ) → B¯ P (ρ) P¡ Proof. Since (ρ) =kerdθ : (ρ)( ) (ρ−1)( ) ,wehavethat (ρ) is a S F c P(1) P¡ weight graded sub-dg- -bimodule of (ρ)(Σ ). It remains to show that is F c P(1) P¡ stable under the coproduct ∆ of (Σ ). Let ξ be an element of (ρ),that’sto ∈Fc P(1) say, ξ (ρ)(Σ )anddθ(ξ)=0.Denote 1 1 2 2 ∆(ξ)= (ξ1 , ..., ξa1 ) σ (ξ1, ..., ξa2 ), Ξ where the sum is on a family Ξ of 2-level graphs, with ξj ∈Fc (ΣP(1)). The fact i (sij ) that dθ is a coderivation means that dθ ◦ ∆(ξ)=∆◦ dθ(ξ). Therefore, we have

a1 ◦ ± 1 1 1 2 2 dθ ∆(ξ)= (ξ1 , ..., dθ(ξk), ..., ξa1 ) σ (ξ1 , ..., ξa2 ) Ξ k=1 a2 ± 1 1 2 2 2 + (ξ1, ..., ξa1 ) σ (ξ1 , ..., dθ(ξk) ..., ξa2 ) =0, k=1 where the d (ξj)belongtoF c (Σ(P(1) ⊕P(2))) with only one vertex indexed θ i (sij −1) P(2) j by Σ .Byidentiﬁcation,wehavedθ(ξi ) = 0 for every i, j.

In the same way, one can see that the Koszul dual of C is a quotient of F(Σ−1C(1)). It remains to show that the product µ on F(Σ−1C(1)) deﬁnes a product on the quotient. We consider the product 1 1 2 2 µ (c1, ..., dθ (c), ..., ca1 ) σ (c1, ..., ca2 ) , j F −1C(1) F −1 C(1) ⊕C(2) where the ci belong to (sij )(Σ )andthec to (s)(Σ ( )) with C(2) j F −1C(1) one vertex indexed by .Sincetheci are elements of (sij )(Σ ), we have j dθ (c ) = 0. The fact that dθ is a derivation gives i 1 1 2 2 µ (c , ..., dθ (c), ..., c ) σ (c , ..., c ) 1 a1 1 a2 1 1 2 2 = dθ µ (c1, ..., c,..., ca ) σ (c1, ..., ca ) . 1 2 1 1 2 2 Therefore the product µ (c1, ..., dθ (c), ..., ca1 ) σ (c1, ..., ca2 ) is null on the cokernel of dθ . 7.1.3. Quadratic properad. We show here that the dual construction is a quadratic construction.

We recall from 2.9 that a quadratical properad is a properad of the form F(V )/ (R)whereR belongs to F(2)(V ). If we consider the weight given by the number of indexed vertices, the ideal (R) is homogenous and the quadratic properad is weight graded.

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Every quadratic properad is completely determined by its graduation P = P(ρ), ρ∈N

the dg-S-bimodule P(1) and the dg-S-bimodule P(2). One has V = P(1) and R = µ ker F(2)(P(1)) −→P(2) . Lemma 7.3. Let C be a weight graded connected dg-coproperad. The Koszul dual C¡ is a quadratic properad generated by C¡ −1C(1) C¡ −1C(2) →F −1C(1) (1) =Σ and (2) =coker θ :Σ (2)(Σ ) . F C¡ →C¡ C¡ Proof. Denote R =ker (2)( (1)) (2) =Im(θΣ−1C(2) ). By deﬁnition, (ρ) − F 1C(1) B¯c C(ρ) is the quotient of (ρ)(Σ ) by the image of dθ on (ρ−1)( ). This image corresponds to graphs with ρ vertices such that at least one pair of adjacent vertices is the image of an element of Σ−1C(2) under θ. This is isomorphic to the part of weight ρ of the ideal generated by R. Therefore, we have C¡ = F(Σ−1C(1))/(R).

Remark. Dually, one can deﬁne the notion of quadratic coproperad and prove that the Koszul dual of a properad is a quadratic coproperad. 7.2. Koszul properads and Koszul resolutions. Here we give the notions of Koszul properad and coproperad. When a properad P is a Koszul properad, it is quadratic, its dual is Koszul and its bidual is isomorphic to P.Weprovethat a properad P is Koszul if and only if the reduced cobar construction of P¡ is a resolution of P.

7.2.1. Definitions. Definition (Koszul properad). Let P be a weight graded connected dg-proper ad. The properad P is a Koszul properad if the natural inclusion P¡ → B¯(P)isa quasi-isomorphism. In the same way, we have the definition of a Koszul coproperad. Definition (Koszul coproperad). Let C be a weight graded connected dg-coproperad. We call C a Koszul coproperad if the natural projection B¯c(C) C¡ is a quasi- isomorphism. Proposition 7.4. If P is a weight graded Koszul properad, then its Koszul dual ¡ P¡ is a Koszul coproperad and P¡ = P.

Proof. The properad P is concentrated in (homological) degree 0 (δP =0and P P P¡ S 0 = ). In this case, (ρ) is a homogenous -bimodule of degree ρ and the B¯c P¡ (ρ) B¯c P¡ (ρ) elements of degree 0 of ( ) correspond to the elements of (ρ)( ) .This shows that ¡¡ B¯c P¡ (ρ) B¯c P¡ (ρ) P (ρ) H0 (ρ)( ) = Hρ ∗( ) ,dθ = . Since the properad P is Koszul, the inclusion P¡ → B¯(P) is a quasi-isomorphism. Using the comparison lemma for quasi-free properads (Theorem 5.9), we show that the induced morphism B¯c(P¡) → B¯c(B¯(P)) is a quasi-isomorphism. Since the bar- cobar construction is a resolution of P (Theorem 5.8), the cobar construction B¯c(P¡)

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is quasi-isomorphic to P.TheS-bimodule P is homogenous of degree 0, therefore we have (ρ) c ¡ (ρ) P if ∗ =0, H∗ B¯ (P ) = 0otherwise. ¡ Finally, we get that P¡ is a Koszul coproperad and that P¡ = P. Corollary 7.5. Let P be a weight graded connected properad. If P is a Koszul properad, then P is quadratic. Proof. If P is a Koszul properad, by the preceding proposition, we know that ¡ ¡ P = P¡ . Also, Lemma 7.3 shows that P¡ is quadratic. 7.2.2. Koszul resolution and minimal model. We have seen in the proof of the previous proposition that, when P is a Koszul properad, the quasi-isomorphism P¡ → B¯(P) induces a quasi-isomorphism B¯c(P¡) → B¯c(B¯(P)). When we compose this quasi-isomorphism with the bar-cobar resolution ζ : B¯c(B¯(P)) →P(cf. Theorem 5.8), we get a quasi-isomorphism of weight graded dg-properads: B¯c(P¡) →P. Deﬁnition (Koszul resolution). When P is a Koszul properad, the resolution B¯c(P¡) →P is called a Koszul resolution. Moreover, we have the following equivalence. Theorem 7.6 (Koszul resolution). Let P be a weight graded connected dg-properad. The properad P is a Koszul properad if and only if the morphism of weight graded dg-properads B¯c(P¡) →P is a quasi-isomorphism. Proof. On the other hand, if the morphism B¯c(P¡) →Pis a quasi-isomorphism, then B¯c(P¡) → B¯c(B¯(P)) is also a quasi-isomorphism. Since we work with weight graded quasi-free properads, we can apply the comparison lemma for quasi-free properads (Theorem 5.9) which gives that the inclusion P¡ → B¯(P)isaquasi- isomorphism. When P is a Koszul properad concentrated in degree 0, we have a resolution ¡ B¯c(P¡)=F c(Σ−1P ) →Pbuilt on a quasi-free properad and such that the diﬀer- −1 ¡ −1 ¡ ential is quadratic dθ (Σ P ) ⊂F(2)(Σ P ). By analogy with associative algebras and operads, we call this resolution the minimal model of P.

7.3. Koszul complex. It is as diﬃcult to show that a properad P is a Koszul properad from the deﬁnition (P¡ → B¯(P) quasi-isomorphism) as to show that the cobar construction of P¡ is a resolution of P (B¯c(P¡) →Pquasi-isomorphism). We introduce a smaller chain complex whose acyclicity is a criterion which proves that P is a Koszul properad.

7.3.1. Koszul complex with coefficients. Following the same method as in the definition of the bar construction with coefficients (cf. 4.2.2), here we introduce the notion of a Koszul complex with coefficients. Definition (Koszul complex with coefficients). Let P be a weight graded connected dg-properad and let L and R be two modules (on the right and on the left) on P.We call a Koszul complex with coefficients in the modules L and R, the chain complex

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¡ deﬁned on the S-bimodule L c P c R by the diﬀerential d, the sum of the three following terms :

(1) the canonical differentials δP , δR and δL induced by the ones of P,R and L. − (2) The homogenous morphism dθL of degree 1 that comes from the structure of P-comodule on the left on P¡ : P¡ −∆→P¡ P¡ ⊕ P¡ P¡ → ⊕P(1) P¡ θl : c (I ) c (I ) c . 1 1 − (3) The homogenous morphism dθR of degree 1 that comes from the structure of P-comodule on the right on P¡ : P¡ −∆→P¡ P¡ P¡ ⊕ P¡ →P¡ ⊕P(1) θr : c c (I ) c (I ). 1 1 We denote it K(L, P,R). Since P¡ can be injected into the bar construction B¯(P), the Koszul complex with coefficients is a sub-S-bimodule of the bar construction with coefficients. Proposition 7.7. Let P be a weight graded connected dg-properad and let L and R be two modules (on the right and on the left) on P. The dg-S-bimodule K(L, P,R)= ¡ Lc P c R is a sub-complex of the bar construction with coefficients B(L, P,R)= L c B¯(P)c R. Proof. This proposition relies on the fact that the differential of the bar construction is defined with the coproduct ∆ of B¯(P)=F c(ΣP) and that the differential of the Koszul complex is also defined with the coproduct on P¡. The Koszul dual P¡ is a sub-coproperad of B¯(P), which concludes the proof.

The inclusion P¡ → B¯(P) induces a morphism of dg-S-bimodules ¡ L c P c R→B(L, P,R).

7.3.2. Koszul complex and minimal model. We have studied a particular bar construction, namely the augmented bar construction B(I, P, P)=B¯(P) c P. Here we consider the equivalent chain complex in the framework of Koszul complexes. Deﬁnition (Koszul complex). We call a Koszul complex the chain complex ¡ K(I, P, P)=P c P ¡ (and K(P, P,I)=P c P ).

The differential of this complex is defined by the sum of the morphism dθr with the canonical differential δP induced by the one of P. Remark that the morphism ∈P¡ P¡ dθr corresponds to extract one operation ν (1) of from the top, identify this P P¡ ∼ P P operation ν as an operation of (1) (since (1) = (1)) and finally compose it in . This can be graphically resumed by the same kind of diagram as the ones given by J.-L. Loday in [L1] for associative algebras. The main theorem of this paper is the following criterion.

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Theorem 7.8 (Koszul criterion). Let P be a weight graded connected dg-properad. The following assertions are equivalent: (1) The properad P is a Koszul properad (the inclusion P¡ → B¯(P) is a quasi-isomorphism). ¡ (2) The Koszul complex P c P is acyclic. ¡ (2 ) The Koszul complex P c P is acyclic. (3) The morphism of weight graded dg-properads B¯c(P¡) P is a quasi-isomorphism.

Proof. We have already seen the equivalence (1) ⇐⇒ (3) (cf. Theorem 7.6). By the comparison lemma for quasi-free P-modules (on the right), the assertion ¡ (1) is equivalent to the fact that the morphism P c P→B¯(P) c P is a quasi- isomorphism. The acyclicity of the augmented bar construction (Theorem 4.15) ¡ shows that the properad P is Koszul (1) if and only if the Koszul complex P c P is acyclic (2). We use the comparison lemma for quasi-free P-modules on the left, to prove the equivalence (1) ⇐⇒ (2).

7.4. Koszul duality for props. One can write the same theory for quadratic props. We are going to show how to retrieve Koszul duality for quadratic props deﬁned by connected relations (cf. 2.9) from Koszul duality of properads.

Let P be quadratic prop F(V )/(R) deﬁned by connected relations R ⊂F(2)(V ) ⊂

F(2)(V ). Recall from Proposition 2.10 that P = F(V )/(R)=S(F(V )/(R)) is isomorphic to the free prop on the properad P = F(V )/(R). With the same methods, one can generalize the bar and cobar constructions, the Koszul duals and complexes to props. Denote by the constructions related to props. Since the concatenation of elements of P is free, the following diagram is commutative: / ¡ F c P¯ P (ΣO )

? P¡ /F c(ΣP¯).

In other words, the Koszul dual coprop P¡ is a coproperad isomorphic to the Koszul dual P¡ of P. One can also see that the Koszul complex P¡P associated to the prop ¡ P is isomorphic to S(P cP). Hence, one is acyclic if and only if the other is acyclic. ¯ The cobar construction F (Σ−1P¡)onP¡ is isomorphic to S(F(Σ−1P¯¡)) = S(B¯c(P¡). Since P = S(P), the cobar construction on P¡ is a resolution of P (morphism of props) if and only if the cobar construction on P¡ is a resolution of P (morphism of properads).

Corollary 7.9. Let P be a quadratic connected dg-prop deﬁned by connected relations. The following assertions are equivalent. (1) The Koszul complex P¡ P is acyclic. ¯ (2) The morphism of weight graded dg-props B c(P¡) → P is a quasi-isomorphism.

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Remark. Since the author does not know any type of gebras deﬁned on a quadratic prop with non-confected relations, we are reluctant to write Koszul duality for props in whole generality. The Koszul resolutions for props are based on the free prop which is much bigger than the free properad. For the applications of Koszul duality studied in this paper (for instance notion of P-gebra up to homotopy cf. 8.4), the resolution of the underlying properad is enough and contains all the information about the algebraic structure. 7.5. Relation with the Koszul duality theories for associative algebras and for operads. The notion of a Koszul dual of a properad introduced here is a coproperad (cf. 7.1). On the contrary, the Koszul dual of an associative algebra isanalgebrain[Pr],andtheKoszuldualofanoperadisanoperadin[GK].To recover these classical constructions, we consider their linear dual : the Czech dual of P¡.

Definition (Czech dual of an S-bimodule). LetP be an S-bimodule; we define the P∨ P S P∨ P∨ Czech dual of by the -bimodule := ρ, m, n (ρ)(m, n), where P∨ ⊗ P ∗ ⊗ (ρ)(m, n):=sgnSm k (ρ)(m, n) k sgnSn . The Czech dual is the linear dual twisted by the signature representations. Lemma 7.10. Let (C, ∆,ε) be a weight graded coproperad such that the dimensions of the modules C(ρ)(m, n) are finite over k, for every m, n and ρ.TheS-bimodule C∨ is naturally endowed with a structure of a weight graded properad. ∨ Proof. Denote P = C . The coproduct ∆ can be decomposed by the weight ∆ = C∨ ρ∈N ∆(ρ). To define the multiplication µ on , we dualize

∆(ρ)(m, n):C(ρ)(m, n) → (C c C)(ρ)(m, n). More precisely, we have ⎛ ⎞

using the finite dimensional hypothesis of the C(ρ)(m, n) and identifying the invari- ants with the coinvariants (we work over a field k of characteristic 0). Hence, we define the composition µ(ρ) by the formula : ⎛⎛ ⎞ ⎞ ∨ ⎜⎜ ⎟ ⎟ P P ⎝⎝ C | | | | ⎠ ⎠ ( c )(ρ)(m, n)= (ρν )( Out(ν) , In(ν) ) ∈G2 ν vertex# of g g c (m, n) ≈ ⎛ ⎛ ρν =ρ ⎞ ⎞ ∨ ⎜ ⎜ ⎟ ⎟ −→ ⎝ ⎝ C | | | | ⎠ ⎠ (ρν )( Out(ν) , In(ν) ) ∈G2 ν vertex# of g g c (m, n) ≈ ρν =ρ t C C ∨ −−−∆(ρ→C) ∨ P =( c )(ρ)(m, n) (ρ)(m, n)= (ρ)(m, n).

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The coassociativity of the coproduct ∆ induces the associativity of the product µ. Also, the counit of C ε : C→I gives, after passage to the dual, the unit of P : ε : I →P.

Proposition 7.11. Let P be a weight graded properad (for instance a quadratic properad). Denote V = P(1),theS-bimodule generated by the elements of weight 1 of P. If there exist two integers M and N such that V (m, n)=0for m>Mor n>Nand if the dimension of every module V (m, n) is finite on k, then the cobar construction B¯c(P) on P and the Koszul dual P¡ verify the hypothesis of the preceding lemma. Proof. Since P¡ is a weight graded sub-coproperad of the cobar construction B¯c(P), ¯c it is enough to show that the dimension of the modules B (P)(ρ)(m, n) is finite. B¯c P F c We have seen that ( )(ρ)(m, n)= (ρ)(V )(m, n). In the case where V verifies the hypotheses of the proposition, this module is given by a sum indexed by the set of graphs with ρ vertices and such that each vertex has at most N inputs and M outputs. Since this set is finite, the dimension of the modules V (m, n) is finite.

Corollary 7.12. For every quadratic# properad generated by an S-bimodule V such P¡∨ that the sum of the dimensions m, n dimk V (m, n) is finite, the Czech dual of the Koszul dual of P is endowed with a natural structure of properad . ∨ ∨ Moreover, if P = F(V )/(R), then the properad P¡ is quadratic and P¡ = F(ΣV )/(Σ2R⊥). We follow the classical notation and denote this properad P!. P! P¡ P¡ One can notice that the properad is determined by (1) =ΣV and (2) = Σ2R. Recall that in the case of quadratic algebra A = T (V )/(R), S. Priddy defined the dual algebra A! by T (V ∗)/(R⊥), when V is finite dimensional. In the same way, V. Ginzburg and M. M. Kapranov define the dual of a quadratic operad P = F(V )/(R)byP! = F(V ∨)/(R⊥), when V is finite dimensional.

In the finite dimensional case, the conceptual definitions of the dual objects given here correspond to the classical definitions up to suspension (cf. [BGS] and [Fr]). ∗ ¡ n ! In particular, the dual A(n) of an algebra A is isomorphic to Σ A n, and the dual ∨ P¡ P nP! (n) of an operad is isomorphic to Σ n . When P is an algebra, the Koszul complex and the Koszul resolution, coming from the cobar construction, introduced here correspond to the ones given by [Pr]. When P is an operad, they correspond to the ones given by [GK].

8. Examples and P-gebra up to homotopy The last difficulty is to be able to prove that the Koszul complex of a properad is acyclic. We consider a large class of properads (properads defined by a replacement rule), and we show that they are Koszul properads. The examples of the properad BiLie of Lie bialgebras and the properad εBi of infinitesimal Hopf bialgebra fall into this case. This method is a generalization to properads of works of T. Fox, M. Markl [FM],M.Markl[Ma3]andW.L.Gan[G].WedefinethenotionofP-gebra up to homotopy and give the examples of several P-gebras up to homotopy.

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8.1. Replacement rule. Let P be a quadratic properad of the form P = F(V, W)/(R ⊕ D ⊕ S),

where R ⊂F(2)(V ), S ⊂F(2)(W )and ⊂ ⊕ ⊕ ⊕ ⊕ D (I W ) c (I V ) (I V ) c (I W ). 1 1 1 1 The two pairs of S-bimodules (V, R)and(W, S) generate two properads denoted as A = F(V )/(R)andB = F(W )/(S). Deﬁnition (Replacement rule). Let λ be a morphism of S-bimodules ⊕ ⊕ → ⊕ ⊕ λ :(I W ) c (I V ) (I V ) c (I W ) 1 1 1 1 such that the S-bimodule D is deﬁned by the image − ⊕ ⊕ → ⊕ ⊕ ⊕ ⊕ (id, λ): (I W ) c (I V ) (I W ) c (I V ) (I V ) c (I W ). 1 1 1 1 1 1

We call λ a replacement rule and denote D by Dλ if the two following morphisms are injective: ⎧ A B →P ⎨⎪ c , 1 2 A B →P ⎩⎪ c . 2 1 Remark. The last condition must be seen as a coherence axiom. It ensures that the natural morphism A B→Pis injective (cf. [FM]). The examples treated here are of this form. They come from the same kind of “distributive law” between two operads described in [FM] and in [Ma3]. Deﬁnition (Reversed S-bimodule). Let P be an S-bimodule. We deﬁne its reversed S-bimodule by Pop(m, n):=P(n, m). The action to take the reversed S-bimodule Pop corresponds to inverse the direction of the operations of P.LetP and Q be two S-bimodules; we have op op op (P c Q) = Q c P . When the S-bimodule P is endowed with a structure of properad, the reversed S-bimodule Pop is also a properad. The properads treated (BiLie and εBi) are generated by operations (n inputs and one output) and cooperations (one input and m outputs). One can write them as F(V ⊕ W )/(R ⊕ Dλ ⊕ S), where V represents the generating operations (V (m, n)=0form>1) and where W represents the generating cooperations (W (m, n)=0forn>1). The properad A = F(V )/(R) is an operad, and the properad Bop = F(W op)/(Sop) is an operad too. In this case, the replacement rule has the following form: ⊗ → ⊕ ⊕ λ : W k V (I V ) c (I W ). 1 1

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For the properad BiLie associated to Lie bialgebras (cf. 2.9), one has A = Bop = Lie and the replacement rule λ is given by 12A 12 21 12 21 A}} D D D D D D A → zz Dzz − zz Dzz Dzz D − Dzz D λ : }} A + . 12 12 12 12 12 In the case of the properad εBi of an inﬁnitesimal Hopf bialgebra (cf. 2.9), the operads A and Bop are equal to the operad As of associative algebras, and the replacement rule is given by ? ? ? ? ? ? → ? ?? λ : ? + . A replacement rule allows us to vertically permutate operations and cooperations.

Lemma 8.1. Every properad of the form P = F(V, W)/(R ⊕ Dλ ⊕ S) defined by a replacement rule λ is isomorphic to the S-bimodule ∼ P = A c B. Proof. As we have seen in the previous remark, the last condition defining a replacement rule ensures that the natural morphism A c B→Pis injective. To show that it is also surjective, we choose a representative in F(V ⊕W )ofevery element of P. It can be written as a finite sum of graphs indexed by operations of V and W . For every graph, we fix each vertex on a level and we vertically permutate elements of V and elements of W with the replacement rule to finally have all the elements of V under the ones of W . Therefore, we get that an element of F(V ) c F(W ) which projected in A c B gives the wanted element. In the two previous examples, this lemma shows that we can write every element of P as a sum of elements of A c B. This corresponds to put the cooperations of B at the top and the operations of A under. In the case of Lie bialgebras, this result has been proved by B. Enriquez and P. Etingov in [EE] (Section 6.4). 8.2. Koszul dual of a properad defined by a replacement rule.

Proposition 8.2. Let P be a properad of the form P = F(V, W)/(R ⊕ Dλ ⊕ S) deﬁned by# a replacement rule λ and such that the sum of the dimensions of V and ⊕ W on k, m, n dimk(V W )(m, n), is ﬁnite. The dual properad is given by ! ∨ ∨ 2 ⊥ 2 2 ⊥ P = F(ΣW ⊕ ΣV )/(Σ S ⊕ Σ Dtλ ⊕ Σ R ), where the replacement rule is tλ. Also, the dual coproperad is isomorphic to the S-bimodules ¡ ∼ ¡ ¡ P = B c A . ∨ ∼ Proof. Corollary 7.12 shows that P! = P¡. ⊥ F ∨ ⊥ Denote R as the orthogonal of R in (2)(V ), S the orthogonal of S in F ∨ ⊥ ⊕ ∨ ⊕ ∨ ⊕ ∨ (2)(W )andDλ the orthogonal of Dλ in (I W ) c (I V ) (I V ) c 1 1 1 ∨ (I ⊕ W ). Therefore, the dual properad P! is given by 1 P! F ∨ ⊕ ∨ 2 ⊥ ⊕ 2 ⊥ ⊕ 2 ⊥ = (ΣV ΣW )/(Σ R Σ Dλ Σ S ).

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S ⊥ It remains to see that an -bimodule Dλ is isomorphic to the image of the morphism −t ⊕ ∨ ⊕ ∨ → ⊕ ∨ ⊕ ∨ ⊕ ∨ ⊕ ∨ (id, λ):(I V ) c(I W ) (I V ) c(I W ) (I W ) c(I V ), 1 1 1 1 1 1

⊥ which means that Dλ = Dtλ. ! ! ∼ ! ! Applying the previous lemma to the properad P ,wegetP = B c A and ¡ ∼ ¡ ¡ P = B c A by Czech duality.

The two examples given by the properads BiLie and εBi of Lie bialgebras and infinitesimal Hopf bialgebras verify the hypotheses of the proposition. They are generated by a finite number of operations and cooperations. ? ?? Since the composition of the form ? does not appear in their definition, we must have it in the relations of the Koszul duals.

Corollary 8.3. (1) TheKoszuldualproperadBiLie! of the properad of Lie bialgebras is given by

BiLie! = F(V )/(R),

12 ?? ?? where V = ⊕ , that’s to say, a commutative operation and a cocom- 12 mutative cooperation, and ⎧ ⎛ ⎞ ⎪ 123 123 ⎪ ?? ?? ?? ⎪ ⎝ ?? − ?? ⎠ S ⎪ .k[ 3] ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ ?? ?? ⎪ ⊕ S ⎝ ? ? − ? ⎠ ⎪ k[ 3]. ? ? ? ⎪ ⎨⎪ 123 123 R = ⎪ 12 12 12 21 12 12 12 21 ⎪ ?? ?? ?? ?? ⎪ ?? ?? ?? ?? ?? ?? ⎪ ⊕ ?? − ⊕ ?? − ⊕ ?? − ⊕ ?? − ⎪ ⎪ ⎪ 12 12 12 12 12 12 12 12 ⎪ ⎪ ⎪ ?? ⎪ ? ⎩ ⊕ ? .

It corresponds to the dioperad of non-unitary Frobenius algebras. As an ! S-bimodule, it is isomorphic to BiLie (m, n)=k, with trivial actions of Sm and Sn. (2) The Koszul dual properad εBi! of the properad of inﬁnitesimal Hopf algebras is given by

εBi! = F(V )/(R),

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A KOSZUL DUALITY FOR PROPS 4937 ?? ?? or V = ⊕ and ⎧ ?? ?? ?? ⎪ ?? ?? ⎪ − ⎪ ⎪ ⎪ ⎪ ⊕ ?? − ?? ⎪ ?? ?? ?? ⎪ ⎪ ⎨ ?? ?? ?? ???? R = ⊕ ??− ⊕ ??− ⎪ ⎪ ⎪ ⎪ ⎪ ?? ?? ?? ?? ⎪ ?? ?? ?? ?? ?? ⎪ ⊕ ?? ⊕ ?? ⊕ ?? ⊕ ?? ⊕ ?? ⎪ ? ? ? ? . ⎩⎪ ? ? ? ?

It is isomorphic, as an S-bimodule, to ! εBi (m, n)=k[Sm] ⊗k k[Sn]. 8.3. Koszul complex of a properad deﬁned by a replacement rule.

Proposition 8.4. Let P be a properad of the form P = F(V, W)/(R ⊕ Dλ ⊕ S) defined# by a distributive law λ and such that the sum of the dimensions of V and W ⊕ A B on k, m, n dimk(V W )(m, n), is finite. We define the two properads and by A = F(V )/(R) and B = F(W )/(S). We suppose that B is a properad concentrated in homological degree 0.IfA and B are Koszul properads, then P is a Koszul properad too. Proof. Lemma 8.1 and Proposition 8.2 show that the Koszul complex of P is of the following form: ¡ ¡ ¡ ¡ ¡ P c P =(B c A ) c (A c B)=B c (A c A) c B. ¡ We introduce the following filtration of the Koszul complex Fn(P c P)which ¡ ¡ corresponds to the sub-S-bimodule of B c (A c A) c B generated by 4-level graphs where the sum of the weight of the elements of B¡ on the fourth level is less than n. This filtration is stable under the differential of the Koszul complex. ∗ Therefore, it induces a spectral sequence denoted as Ep, q. The first term of this 0 S B¡ A¡ A B spectral sequence Ep, q is isomorphic to the sub- -bimodule of c ( c ) c composed by elements of homological degree p + q and such that the sum of the weight of the elements of B¡ on the fourth level is equal to p.SinceB is concentrated 0 B¡ A¡ A B in homological degree 0, we have Ep, q = c ( c ) c . The differential p q d0 is the differential of the Koszul complex of A.SinceA is a Koszul properad, we 1 1 ¡ get Ep, q =0ifq =0andEp, 0 = B cB. The differential d1 is the differential of p the Koszul complex of B. Once again, since B is a Koszul properad, we have

I if p = q =0, E2 = p, q 0otherwise. The ﬁltration is exhaustive and bounded from below. Therefore, we can apply the classical theorem of convergence of spectral sequences (cf. [W] 5.5.1) which gives that the spectral sequences converge to the homology of the Koszul complex of P. This complex is acyclic and the properad P is Koszul.

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Corollary 8.5. The properads BiLie of Lie bialgebras and εBi of inﬁnitesimal Hopf algebras. Proof. In the case of BiLie, the properad A is the Koszul operad of Lie algebras Lie and the properad B is the reversed properad of the Koszul operad Lie, B = Lieop, which is also a Koszul properad. InthecaseofεBi, the properad A is the Koszul operad As of associative algebras and the properad B is the reversed properad of the Koszul operad As, B = Asop, which is also a Koszul properad.

Application: The cobar construction of the coproperad BiLie¡ is a resolution of the properad BiLie. We can interpret this homological result in terms of graph cohomology. Since the Koszul dual properad of BiLie is a dioperad, the differential of the cobar construction of BiLie¡ is equal to the boundary map defined by M. Kontsevich in the context of graph cohomology (vertex expansion). Therefore, we find the results of M. Markl and A.A. Voronov in [MV] : the cohomology of the directed connected commutative graphs is equal to the properad BiLie,andthe cohomology of the directed connected ribbon graphs is equal to the properad εBi.

8.4. P-gebra up to homotopy. Deﬁnition (P-gebra up to homotopy). When P is a Koszul properad, we call P- gebra up to homotopy any gebra on B¯c(P¡). We denote the category of P-gebras up to homotopy by P∞-gebras. This notion generalizes to gebras the notion of algebra up to homotopy (on an operad). For instance, we have the notion of Lie bialgebras up to homotopy and the notion of inﬁnitesimal Hopf algebras up to homotopy.

9. Poincareseries´ Here we generalize the Poincar´e series of quadratic associative algebras (cf. J.-L. Loday [L1]) and binary quadratic operads (cf. V. Ginzburg and M.M. Kapranov [GK]) to properads. When a properad P is Koszul, we show a functional equation between the Poincar´eseriesofP and P¡.

9.1. Poincar´e series for Koszul properads. Let P be a weight graded con- ¡ nected dg-properad. Denote by K its Koszul complex K(I, P, P)=P c P.This complex is the directed sum of its sub-complexes indexed by the global weight K K K = m, n, d ≥0 (d)(m, n), where (d)(m, n) has the following form : →P¡ →P¡ P →···→ P¡ P 0 (d)(m, n) c (m, n) c (m, n) (d−1) (1) (1) (d−1)

→P(d)(m, n) → 0. The main theorem of this paper (Theorem 7.8) asserts that a properad P is Koszul if and only if each chain complex K(d)(m, n) is acyclic for d>0. In this case, we know from Corollary 7.5 that the properad P is quadratic. Suppose that P is a quadratic properad generated by an S-module V such that the total dimension of m, n∈N∗ V (m, n) is ﬁnite. In Proposition 7.11 we have seen that each module

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F(d)(V )(m, n) has finite dimension. Therefore, the dimension of each P(d)(m, n) P¡ K and (d)(m, n) is finite. The Euler-Poincaré characteristic of (d)(m, n)isnull

d − k P¡ P ( 1) dim( c )(m, n)=0. k=0 (k) (d−k) The Euler-Poincaré characteristic is given by the following formula: d − k P¡ P ( 1) dim( c )(m, n) k=0 (k) (d−k) Sk,¯ ¯j n! m! P! P¡ = c dim (l1,k1) ...dim (lb,kb) ¯ı!¯j! k¯! ¯l! (o1) (ob) Ξ × P P dim (q1)(j1,i1) ...dim (qa)(ja,ia), where the sum Ξ runs over the n-tuples ¯ı,¯j, k¯, ¯l,¯o andq ¯ such that |¯ı| = n, |j¯| = |k¯|, |¯l| = m, |o¯| = k and |q¯| = d − k. Definition (Poincaré series associated to an S-bimodule). To a weight graded S- bimodule P, we associate the Poincaréseriesdefined by P dim (d)(m, n) m n d fP (y, x, z):= y x z , m! n! m,n≥1 d≥0

when the dimension of every k-modules P(d)(m, n) is ﬁnite. We deﬁne the function Ψ by the formula

b a kβ jα k,¯ ¯j 1 ∂ g 1 ∂ f Ψ g(y, X, z),f(Y, x, z ) := Sc k (y, 0,z) j (0,x,z), kβ! β jα! α Ξ β=1 ∂X α=1 ∂Y where the sum Ξ runs over the n-tuples k¯ andj ¯ such that |k¯| = |j¯|. Theorem 9.1. Every Koszul properad P generated by an S-module of global ﬁnite dimension veriﬁes the following equation: Ψ fP¡ (y, X, −z),fP (Y, x, z) = xy. Proof. We have Ψ fP¡ (y, X, −z),fP (Y, x, z) b k a ∂ fP¡ jα k,¯ ¯j 1 β − 1 ∂ fP = Sc k (y, 0, z) j (0,x,z) kβ! β jα! α Ξ β=1 ∂X α=1 ∂Y d b ¡ a dimP (lβ,kβ) dimP (j ,i ) × − k k,¯ ¯j qβ oα α α m n d ( 1) Sc y x z lβ! kβ! jα! iα! m, n≥1 k=0 Ξ β=1 α=1 d≥0 =0 for d>0 = xy.

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Remark. Since the Koszul complex K(I, P, P) is acyclic if and only if the Koszul complex K(P, P,I) is acyclic, we also have the symmetric relation Ψ fP (y, X, −z),fP¡ (Y, x, z) = xy.

9.2. Poincaré series for associative Koszul algebras. When# the quadratic pr- P d operad is a quadratic algebra A,itsPoincaréseriesequals d≥0 dim(A(d))z xy. If we denote d fA(z):= dim(A(d))z , d≥0 then the functional equation of the previous theorem is equivalent to the well known relation (cf. [L1]) fA(x).fA¡ (−x)=1. 9.3. Poincaré series for Koszul operads. When the quadratic properad P is a quadratic operad, its Poincaréseriesisequalto dimP (n) (d) xn yzd. n! d≥0,n≥1 We denote P dim (d)(n) n d fP (x, z):= x z . n! d≥0,n≥1 Corollary 9.2. The Poincaré series of a Koszul operad P verifies the relation

fP¡ (fP (x, z), −z)=x.

In the binary case (when V = V (2)), we have P(n−1) = P(n)and P dim (n) n n−1 fP (x, z)= x z . n! n≥1 We denote P n dim (n) n fP (x):= (−1) x , n! n≥1 and if P is Koszul, we have that the preceding theorem gives the well known relation (cf. [L2] Appendix B) fP¡ (fP (x)) = x. Example. P F V S V ?? ?? Considerthefreeoperad = ( ) generated by the -module = k{ , , ...} composed by one n-ary operation for every n. Since it is a free operad, it is a Koszul operad. Whereas the total dimension of V is infinite, the dimension of every F(V )(n) is finite. Therefore, we can consider the Euler-Poincaré characteristic of the Koszul complex of P.SinceP¡ = k ⊕ V ,wehave 2 ¡ n d n x fP¡ (x, z)= dimP (n)x z = x + x z = x + z . (d) 1 − x d≥0,n≥1 n≥2 Also, Corollary 9.2 gives the following equation: 2 (z +1)fP (x, z) − (1 + x)fP (x, z)+x =0.

Let Pn(z)bethePoincar´e polynomial of the Stasheﬀ polytope of dimension n n, also called the associahedra# and denoted as K or Kn+2. This polynomial is n n k n equal to Pn(z):= k=0 Celk .z ,whereCelk represents the set of the cells of

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A KOSZUL DUALITY FOR PROPS 4941 # n dimension k of the polytope of dimension n.DenotefK (x, z)= n≥0 Pn(z)x as the generating series. The cells of dimension k of Kn are indexed by planar trees with n +2 leavesand − n P n +1 k vertices. This bijection implies that Celk = dim n+1−k(n +2),which gives n−1 Pn k n fP (x, z)=x + dim k z x n≥2 k=1 n−1 l−2 k n = x + Celn−2−(k−1) z x n≥2 k=1 n 2 n k n = x + zx Celn−k z x n≥0 k=0

1 1 = x + zx2 P (xz)n = x + zx2f xz, . n y K z n≥0

Therefore, we get the following relation verified by fK : 2 2 − ((1 + z)x )fK (x, z)+( 1+(2+z)x)fK (x, z)+1=0. Finally, the generating series of the Stasheff polytopes verifies 1+(2+z)x − 1 − 2(2 + z)x + z2x2 f (x, z)= . K 2(1 + z)x2

Acknowledgements This paper is part of the author’s Ph.D. thesis. I would like to thank my advisor Jean-Louis Loday for his conﬁdence. I also wish to thank Benoit Fresse for his help and for the proof of Theorem 1. I am grateful to Birgit Richter and Martin Markl who have carefully read my thesis and to Wee Liang Gan who pointed out some inaccuracies.

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Laboratoire J. A. Dieudonne,´ Universite´ de Nice, Parc Valrose, 06108 Nice Cedex 02, France E-mail address: [email protected] URL: http://math.unice.fr/∼brunov

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